Fourier and Hadamard Transform Methods in Spectroscopy

A thousand points of light: the Hadamard transform in chemical analysis and ... Resolution and signal-to-noise in Fourier transform mass spectrometry...
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Alan G. Marshall and Melvin B. Comisarow Department of Chemistry University of British Columbia Vancouver, Canada V6T 1W5

Fourier and Hadamard Transform Methods in Spectroscopy To understand why Hadamard and Fourier methods have proved so valu­ able in spectroscopy, it is first neces­ sary to recognize the disadvantages of the conventional way of doing spec­ troscopy, in which the spectrum is ob­ tained by scanning across the spectro­ scopic region of interest with a narrow observation window. Surprisingly, the basic explanation may be expressed very simply, by analogy to the best way to use an ordinary double-pan balance. Weights on a Balance

Consider the common problem of determining the weights of four un­ known objects, by use of the schematic balance shown in Figure 1. Conven­ tionally, we would solve the problem by weighing the unknowns one at a time in (say) the left pan, by putting the appropriate (known) weights on the right, as shown schematically below: Measure­ ment rl r2 =3 =4 No. of weighings

=1

Unknown s2 s3

#4 0 0 0 1

The obvious advantage of this proce­ dure is that the measurements yield each unknown weight directly; there­ fore, no data reduction is required. However, the disadvantage is that each unknown has been weighed only once. In any experimental measurement characterized by a certain level of im­ precision or random noise, it is desir­ able to repeat the measurement many times to obtain a more accurate result. The signal (in this case, the weight of a given unknown) will accumulate as the number of weighings, N. But if the noise is random its magnitude may be treated as a random walk about zero (the average noise level), and the aver­ age absolute distance away from zero

fc»Ii Figure 1. Schematic diagram of double-pan balance, set of standard weights, and four unknown weights

after Ν steps of a random walk (more precisely, the root-mean-square dis­ tance) is proportional to Λ/Ν (1). Thus, the true measure of precision of the repeated measurement, the signalto-noise ratio, is proportional to (N/ VN) or just VN. Let us return to the balance prob­ lem. Suppose that two unknown weights are placed on the left pan at • once and that four linearly indepen­ dent arrangements of unknowns are weighed, two at a time: Measurement si s2 S3 14 No. of weighings

Unknown il #2 #3 1 1 0 0 1 1 0 0 1 1 0 0 2 2 2

if4 0 0 1 1 2

This time, the four unknown weights are related to the four observed weights by four linear algebraic equa­ tions, which may then be solved to yield the desired unknown weights. However, since each unknown has now been weighed twice, the precision (sig­ nal-to-noise ratio) of each calculated weight is now better by V 2 than for the original conventional experiment.

Since the same total number of weigh­ ings (four) are required, the total time required for the new experiment is also the same. For an arbitrary num­ ber, N, of unknown weights, the gen­ eral improvement in signal-to-noise ratio for this encoding-decoding scheme is ViV/2 for an experiment that takes no longer to carry out than Ν conventional single-object weigh­ ings. [The precision of the calculated weight will be better only if the aver­ age error in a particular weighing de­ pends upon the balance and is inde­ pendent of the magnitude of the mea­ sured weight. This general condition is called "detector-limited" noise and is distinguished from the "source-limit­ ed" noise situation in which the rms noise is proportional to the square root of the signal magnitude. If the noise is source limited, the above en­ coding-decoding method will not im­ prove the precision of the calculated weights over those determined in a simpler one-at-a-time weighing proce­ dure (2). Spectroscopic examples in which the noise is detector limited in­ clude infrared, microwave, nuclear magnetic resonance, and ion cyclotron resonance experiments. Examples in which the noise is source limited in­ clude optical (VIS-UV) and charged-

ANALYTICAL CHEMISTRY, VOL. 47, NO. 4, APRIL 1975 · 491 A

particle (photoelectron, ESCA, elec­ tron impact) spectroscopy.] Although the experimental details of the spec­ troscopy experiment are completely different from those of the balance ex­ periment, the preceding encoding-de­ coding scheme still applies and forms the basis for Hadamard transform spectroscopy, as will shortly be evi­ dent. By logical extension of the preced­ ing argument, one might think of put­ ting unknown weights on both sides of the balance rather than just on one side, while keeping track of the (known) weight required to balance any particular arrangement of un­ knowns: MeasureUnknowns in ment Left pan Right pan #1 #1 #2, #3, #4 #2 #1, #2 #3, #4 #3 il, #3 #2, #4 #4 #1, #4 #2, #3 (Each unknown weighed four times) Again, it is possible to extract the four desired individual unknown weights by straightforward solution of four coupled linear algebraic equations. Since each unknown has now been weighed four times, the signal-to-noise ratio for each calculated weight is im­ proved by Λ/4" = 2 times over conven­ tional one-at-a-time weighing. For an arbitrary number of unknowns, N, it follows that the general improvement in signal-to-noise ratio will be a full v W . This improvement also applies to the (experimentally different) Fou­ rier transform spectroscopy experi­ ment, as discussed later in this paper. Hadamard and Fourier methods, then, provide a means for improving the precision (signal-to-noise ratio) in a weighing experiment by a factor of ViV/2 or v W , respectively, but re­ quire the same total time for measure­ ment. This improvement is known as the Fellgett advantage (3); all that re­ mains is to show that we ordinarily perform spectroscopy as inefficiently as we ordinarily use a double-pan bal­ ance, and we then explore the avail­ able means for exploiting the potential Fellgett advantage in the situation. [The Fellgett advantage can be real­ ized only if the noise is detector limit­ ed, not when the noise is source limited (2).] Direct Multichannel Spectrometers Figure 2 is a highly schematic di­ agram of a generalized spectrometer. The dispersive element might be a prism or grating (infrared, optical), for example; the slit might be a band-pass filter for a low-frequency (microwave, nuclear magnetic resonance) case. The slit width is chosen sufficiently narrow

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Figure 2. Top: schematic diagram of single-slit scanning absorption spectrometer. Single-slit scanning emission spectrometer lacks only broad-band source. Bottom: detector readings from number of individual slit positions

MULTICHANNEL DETECTOR

WINDOW Ν

CHANNELS

Figure 3. Schematic diagram of detector section of direct multichannel spectrome­ ter composed of many separate single-channel detectors

that when detector readings are col­ lected from a number of individual slit positions (bottom of Figure 2), there is sufficient resolution to distinguish spectral features of interest. The most important feature of such a spectrom­ eter is that its detection of an absorp­ tion spectrum requires a procedure formally identical to the one-at-a-time method of determining the weights (spectral intensities) of Ν different unknown objects (spectral slit posi­ tions). It would thus be desirable to open up the slit aperture to the full width of the desired spectral window by using Ν separate single-channel detectors as shown in Figure 3. Since all slit positions are now monitored at once, rather than just one at a time, the spectrometer of Figure 3 offers (in principle) an improvement of the full Λ/ÏV advantage in signal-to-noise ratio, compared to the result of a single complete spectral scan requiring the same total time by the spectrometer of Figure 2. Alternatively, it would

492 A · ANALYTICAL CHEMISTRY, VOL. 47, NO. 4, APRIL 1975

be possible to obtain a spectrum having the same signal-to-noise ratio in (l/iV) the time required to scan the Ν individual slit positions one at a time. Because of the conceptual simplici­ ty of the spectrometer of Figure 3, it is logical to investigate its feasibility. It is desirable to be able to resolve spec­ tral detail as narrow as the width of a typical spectral absorption line; there­ fore, the minimum number of chan­ nels that will be required in a multi­ channel spectrometer is simply the width of the entire spectral range of interest, divided by the width of a sin­ gle spectral line. The resultant neces­ sary number of channels for various forms of spectroscopy is shown in Table I. From Table I, it would appear that electronic (VIS-UV) spectroscopy is the least likely candidate for success with a direct multichannel spectrome­ ter; but in fact, multichannel detec­ tion of optical-UV radiation is readily accomplished photographically. The

Table I. M i n i m u m Number of Channels Required for Various Types of Direct Multichannel S p e c t r o m e t e r s Type of spectroscopy

Mossbauer ESCA Photoelectron Electronic Vibrational Rotational '»C NMR ICR

Largest usual frequency

6X 3.5 X 5X 1.5 X 2X 4X 8X 2X

10" Hz 10" l(l·' 10'·' 10': 10'" 10· 10'·

Typical spectral range

Width of 1 line

10" H 7 10' 3 X 10'· 1.2 Χ ΙΟ­ Ι.5 Χ 10" 3 Χ 10"': 2 Χ 10 2 Χ 10'·

10-Hz 10" 10'= 10' 3 Χ 10» 10' 0.5 10Ί

Mm no. of

channels'

10 1,000

3,noo

1,250.000 50,000 300,000 40.000 20,000

'' Nunibor of channels is oblisini'd by dividm(> th« typical spectral rani'.o by the width of une line.

resolution of a fine grain photographic plate is sufficient to provide for the huge number of required channels, since the desired spectrum may be dispersed over the necessary distance (a few meters) without undue effort. In ESCA (electron spectroscopy for chemical analysis) (4) and photoelec­ tron spectroscopy (5), electrons are dislodged from atoms or molecules by X-ray or UV radiation, respectively, and the electrons released have a translational energy which depends on the energy of the bound state occupied by that electron in the original atom or molecule. By scanning the energy of the observed dislodged electrons, the energies of the original molecular elec­ tronic states can be determined. By passing the electrons between two charged parallel plates, the dislodged electrons may be dispersed in space, according to their velocity, to achieve the arrangement shown in Figure 3. This multichannel electron detection scheme has recently become feasible with the advent of the vidicon detec­ tor (6), in which an arriving electron strikes a fluorescent screen on the face of a television camera. Since electrons of different velocity can be dispersed to strike different regions of the screen, their arrival will be recorded independently by different elements of the television camera grid. Because of the small required number of detec­ tor channels (see Table I), the Figure 3 spectrometer is thus now feasible for ESCA and photoelectron spectrosco­ pyWith the other forms of spectrosco­ py listed in Table I, direct multichan­ nel methods are less attractive. For microwave (rotational) spectroscopy, for example, there is no broad-band radiation source available: a blackbody radiation source, such as em­ ployed for other radiation energies (xenon or hydrogen discharge for UV, hot tungsten wire for visible, globar for near infrared and infrared, mercu­ ry vapor for far infrared) would have to be operated at an unreasonably high temperature to obtain sufficient

radiation flux for use as a radiation source. It would be conceivable to con­ struct an array of individual (narrow­ band) microwave transmitters (about $5,000 each) as the "broad-band" ra­ diation source, but Table I shows that the cost would be excessive ($109 . . . ! ) . For infrared spectroscopy, on the other hand, the necessary broad­ band source is available, but it would be necessary to disperse the spectrum over many meters to be able to resolve the desired spectral detail with exist­ ing individual (thermopile) detectors of about 1-mm width each. At a cost of about $200 per detector, the total cost again becomes unmanageable. (Photo­ graphic detection does not extend be­ yond about 12,000 Â and is thus unavailable.) Finally, for nuclear magnetic resonance (NMR) and ion cyclotron resonance (ICR) spectroscopy, broad-band sources are again available, but the cost of an array of tens of thousands of individual narrow-band mixer-filter detectors (see below) is again unreasonably high. For infrared, microwave, and radiofrequency spectrometers, then, the

direct multichannel approach is just not feasible, either geometrically or financially. We will now consider two recent and valuable indirect approaches: Hadamard transform spectroscopy and Fourier transform spectroscopy. Hadamard Transform Encoding-Decoding ("Multiplex") Spectrometers

Figure 4 shows the instrumental modification which allows for use of the Hadamard scheme: by use of the original (inexpensive) broad-band detector of the spectrometer of Figure 2, a mask is interposed between the desired spectroscopic "window" and the detector. The mask is constructed so that its smallest opening is the same as the (narrow) slit width of the conventional spectrometer (Figure 2) but with approximately half the total possible slit positions open. The pattern of open and shut slits is random. Let the spectrum of transmitted intensities (bottom of Figure 2) be represented by spectral elements: x\, x% . . . , XN- When the mask in Figure 4 is in position, the detector total response, y, is composed of a sum of all the desired spectral elements, each weighted by a factor, an, of either zero or one, depending on whether that particular slit was shut or open, respectively: y = fljATj +

azx2 α

+

. . .

+

χ

Ν Ν> α„ = 0 or 1

(1)

In other words, the detector has pro­ vided one observable (y), expressed in terms of Ν unknowns {x\ to XJV), ac­ cording to the "code" (αϊ to αχ) of Equation 1. This situation is clearly parallel to that of putting half the un-

BROAD-BAND DETECTOR

MASK Figure 4. Disperser, mask, and (single, broad-band) detector of Hadamard spec­ trometer

494 A · ANALYTICAL CHEMISTRY, VOL. 47, NO. 4, APRIL 1975

1 0 0 11 0 0

10 01

:

4

0 0 11

1 ι

1

MASKS

0 1 10

- .

1 MASK

1 10 0

Figure 5. Functional equivalence of four separate m a s k s (Λ/ possible slits each) and movable single mask [(2 Ν — 1) possible slits]

known weights on the left pan of the balance, as already discussed. To re­ cover the desired spectrum, x\ to XN, the next step is to remove the first mask and introduce a second mask, again with a random arrangement of open and shut slits with approximate­ ly half the slits open, so that this sec­ ond slit arrangement is linearly inde­ pendent from the first. Proceeding in this way, one readily obtains Ν ob­ servables (the total transmitted inten­ sity through each of Ν different masks, y ι to yjv), expressed in terms of Ν unknowns (the spectrum, χ ι to XN), according to a "code" in which all the coefficients are "either zero or one, and roughly half the coefficients in any one row are zero: •+ a

VN

2\x\

— aN\x\

•*• al2x2 +

a

N2x2

~*~ *

'+

+

•+

a,„xK a

2NXN aanxN (2)

A particularly convenient method for providing the Ν linearly indepen­ dent slit arrangements is shown in Figure 5. Instead of Ν separate masks, it suffices to construct just a single mask consisting of (2 Ν — 1) poten­ tially open slit positions: then by opening the window over just the first Ν positions, one constructs the first slit arrangement at the left of Figure 5; by opening the window over the sec­ ond through (N + 1) positions, one creates the second slit arrangement at the left of the figure, etc. Mechanical­ ly, the change from one slit arrange­ ment to another simply consists of translating the linear (2 Ν — l)-slit mask across the window by one posi­ tion per change. On both computational and me­ chanical grounds, the Hadamard ap­ proach of Figures 4 and 5 conveniently achieves an improvement of a factor of VN/2 in signal-to-noise ratio over the conventional one-slit-at-a-time scan­ ning spectrometer of Figure 2, because half the Ν possible slits are open dur­ ing each measurement, rather than

just one, while both experiments re­ quire the same length of time for exe­ cution of Ν separate intensity mea­ surements. [In the procedure of Figure 5, each successive slit arrangement or mask differs from the immediately preceding one by cyclic permutation of the slit pattern. Successful solution of a given set of simultaneous linear algebraic equations, such as Equations 2, by a digital computer requires that the matrix of the coefficients be "well conditioned" (7). Fortunately, the ma­ trix of coefficients generated by the scheme of Figure 5 can readily be transformed to such a well-condi­ tioned form, and Equations 2 can thus be solved accurately.] Alternatively, the Hadamard spectrometer can pro­ vide the same signal-to-noise ratio in a factor of (2/JV) as much time as would be required by the conventional sin­ gle-slit spectrometer. Hadamard encoding-decoding methods have been applied most prominently in infrared spectroscopy (8, 9).

Incoherent and Coherent Spectrometers

To proceed to Fourier methods, it is important to understand that the spectrometers discussed up to now (Figures 2-5) can operate with an in­ coherent radiation source; that is, there is no necessary common phase relationship (see below) between the various radiation components issuing from the source. For such incoherent source spectrometers, Hadamard mask techniques provide a means for effectively opening up the slit width without sacrificing resolution (Figure 6). There are, however, some major advantages (see below) in using a scanning spectrometer having a coher­ ent source; the coherent source makes possible another type of encoding-de­ coding scheme (Fourier) for opening the spectral window while preserving resolution (Figure 6). Finally, one rea­ son that both Hadamard (incoherent source) and Fourier (coherent source) methods can be applied to infrared spectroscopy is that the Michelson in­ terferometer can be thought of as a device which effectively converts inco­ herent to coherent radiation in the present context. A coherent radiation source and co­ herent detector in a spectrometer pro­ vide two important advantages to the spectroscopist. First, since the fre­ quency of the coherent radiation source is easily determined to very high accuracy by use of an electronic counter, the line positions in a spec­ trum may be determined very accu­ rately, simply by measuring the fre­ quency of the source as it is (slowly) scanned over the spectral window.

INCOHERENT

I

NARROW BAND

HADAMARD TRANSFORM

BROAD BAND

NARROW BAND

BROAD! BAND!

FOURIER TRANSFORM

Figure 6. Spectroscopy classified by radiation source and spectrometer bandwidth

496 A · ANALYTICAL CHEMISTRY, VOL. 47, NO. 4, APRIL 1975

Fourier Transform Spectroscopy MONOCHROMATIC TUNABLE COHERENT SOURCE

Fourier transform methods at first seem strange to our intuition, because we are prejudiced by our eyes and ears to analyze our surroundings in the fre­ quency domain—we judge light by its color and sound by its pitch. It is, however, equally useful to analyze ob­ servations in the time domain (Figure 9). The upper left diagram of Figure 9 shows a simple DC (zero frequency) pulse, which is turned on at time zero, and turned off at time, T. Intuition would suggest that the frequency rep­ resentation of such a pulse should consist simply of a signal at zero fre­ quency, but the actual frequency rep­ resentation consists of a signal which is spread over a range of frequencies near zero. By using a shorter pulse (middle of Figure 9), the frequency representation is spread over an even wider range, and in the limit that the DC pulse is made infinitely narrow (bottom of Figure 9), the frequency representation is a completely flat spectrum. [The mathematical corre­ spondence between the time domain and frequency domain diagrams of Figures 9 and 10 is called a (one-sided) Fourier transformation (14).} The diagrams in Figure 9 suggest that the broad-band frequency excita­ tion required for a multichannel or multiplex spectrometer can be gener­ ated by use of a sufficiently narrow electromagnetic radiation pulse. (If the pulse consists of an AC rather than a DC waveform, then the pic­ tures of Figure 9 still apply, except that the frequency representation is now centered at the AC frequency rather than at zero frequency—see top trace of Figure 10.) For NMR, for example, Table I in­ dicates that an excitation bandwidth of about 10 kHz is required—Figure 10 (top trace) indicates that such an

V

Figure 7. Hypothetical infrared laser source spectrometer

Second, coherent radiation permits the implementation of electronic fil­ tering techniques which can make the spectrometer resolution arbitrarily high. Thus, the spectral lineshape de­ termined by a coherent radiation spectrometer can be made characteris­ tic of the sample, by making the in­ strumental broadening arbitrarily small. The basic operation of a coherent source spectrometer is shown in Fig­ ure 7, consisting of a hypothetical in­ frared laser source spectrometer for use in vibrational spectroscopy. The radiation issuing from the source con­ sists of a plane-polarized electric field whose magnitude varies sinusoidally with time. Upon encountering an elec­ tric dipole (i.e., a polar molecule), the electric field will force the dipole to oscillate at the frequency of the radia­ tion, and the amplitude of that dipolar oscillation will be greatest when the electric field oscillation frequency is the same as ("in resonance with") the "natural" vibration frequency of the dipole. If the source radiation is coher­ ent, then all the dipoles in a given re­ gion of space will oscillate together, forming a macroscopic oscillating elec­ tric dipole in the sample. That macro­ scopic oscillating dipole then induces an oscillating charge (and thus a cor­ responding oscillating voltage) on the parallel plates of the capacitor enclos­ ing the sample in Figure 7. That in­ duced oscillating voltage may then be amplified and (in the most important step) multiplied (in a "mixer") by the oscillating signal from the source and the product decomposed electronically into the sum and difference of the two sine wave frequencies, just as the product of two sine waves may be de­ composed algebraically (by a trigono­ metric identity) into sine waves of the sum and difference frequencies. The low-pass filter rejects the (higher) "sum" frequency and passes the (lower) "difference" frequency which is then recorded. The above mixing process effectively extracts a small

spectral segment which is centered at the source frequency and whose width is determined by the bandwidth of the low-pass filter. For example, the tran­ sient ion cyclotron resonance signal in Figure 11 (see below) was obtained by just this sort of mixing and filtering procedure. In more familiar language, this sort of spectrometer provides a slit posi­ tion which is determined by the fre­ quency of the source and a slit width which is determined by the bandwidth of the electrical low-pass filter and which may therefore be made arbi­ trarily wide or narrow without any mechanical adjustment of the spec­ trometer geometry. Spectrometers in which a macroscopic change in a phys­ ical property of the sample is induced by radiation from a coherent source, and that macroscopic change is de­ tected electronically, in the manner described above, have long been em­ ployed in NMR spectroscopy (10) and ion cyclotron resonance (Figure 8) spectroscopy (11) and have recently been introduced in microwave (12) and infrared (13) spectroscopy.

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498 A · ANALYTICAL CHEMISTRY, VOL. 47, NO. 4, APRIL 1975

M" H

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.hand, the oscillation is observed for several lifetimes of its decay (middle trace of Figure 10), the spectral repre­ sentation approaches the familiar Lorentzian line shape encountered in many forms of spectroscopy. Finally, the bottom trace of Figure 10 illus­ trates the intermediate case in which the acquisition time, T, is of the order of the decay lifetime, r. The irrevers­ ible decay of the oscillation is due to radiative damping ("spontaneous emission") (17) and to interactions of the sample (nucleus, ion, molecule) with its surroundings, where the inter­ action may be neutral-neutral colli­ sions (microwave, infrared, optical); ion-molcule collisions (ion cyclotron resonance); rotational diffusion (nu­ clear magnetic resonance, electron spin resonance); or depletion of the excited species owing to chemical reaction. The multichannel advantage of the Fourier approach can now be under­ stood. Suppose that the time domain response, y (t), is sampled at Ν equally spaced intervals during a total acqui­ sition time, T. Each of these sampled time domain points, y(tn), η = 1 to N, is then a linear combination of all the discrete frequency domain spectral points, χ(ωη), according to: yiti)

= αιιχ(ωι)

+ α12χ(ω2) + . . . +

,j.|i,

1

-•--..-400

;

" ""' '

:

s

a·-u', sec Figure 10. Frequency representations (right) of three types of time domain (left) spectrometer signals

excitation may be produced simply by applying a radiofrequency pulse whose duration is of the order of 10 μββο. As another example, electron impact spectroscopy (15) is based on the rapid passage of an electron past a molecule. This passing electron pro­ duces a very short, sharp pulse of elec­ tric field at the molecule and thus acts as a very broad-band, nearly flat source of irradiation. In this case, the frequency bandwidth is sufficient to excite the same sort of transitions as are more conventionally studied in photoelectron and ESCA spectrosco­ py-

y(t2) = αΖχχ(ω1) + α22χ(ω2) + . . . + α2Νχ(ωΝ) : V (tir) = άΝίχ(ωχ) + αΝ2χ(ω2) + . . . + αΝΝχ(ωΝ) (3) in which a nm = e x P t 2 Trimtn/T] (4) o r just anm = exp [2 mmri/N] (5)

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When a given single oscillator is subjected to irradiation at its resonant frequency, the amplitude of oscillation will increase. If the irradiating excita­ tion is then removed, the oscillation will persist with an amplitude which decreases (usually exponentially) with time, as shown for three convenient limiting situations at the left of Figure 10. If the oscillation is not appreciably reduced during the acquisition time, Τ (top trace of Figure 10), then the cor­ responding frequency representation has a functional form which resembles the amplitude of (Fraunhofer) diffrac­ tion by a slit (16). If, on the other

500 A · ANALYTICAL CHEMISTRY, VOL. 47, NO. 4, APRIL 1975

Equations 3 should be compared to Equations 2: since there are now Ν in­ dependent observed sampled time do­ main points, y (ii) to y(tw), each ex­ pressed in terms of all Ν discrete fre­ quency domain points, χ (ωχ) to χ (ωχ), it is again possible to "decode" the ob­ served data to obtain the desired spec­ trum. The decoding procedure, called a discrete Fourier transformation, may be calculated rapidly by a digital computer (18). In contrast to the Hadamard technique, in which half the possible spectrum is detected in any given observation (i.e., half the anm in any one row of Equations 2 are zero), the magnitude of each anm in the Fourier experiment of Equations 3 is unity: = | exp [2 mnm/N] \ (6)

so that in the Fourier experiment, it is as if all the possible slits are open. By the arguments previously used for the double-pan balance example, it is now clear that detection of the time domain response, followed by Fourier transformation to obtain the frequen­ cy domain response, provides a fre­ quency spectrum exhibiting either sig­ nal-to-noise improvement of a factor of vTV in the same total observation period; or a spectrum having the same signal-to-noise ratio in a factor of (1/ N) as much time as required by a con­ ventional spectrometer which scans the spectrum slowly with'a narrowbandwidth detector. [For the unique case of Fourier transform infrared spectrometers based on the Michaelson interferometer, the spectrum is obtained by discrete Fourier transfor­ mation of the (spatially dispersed) sampled interferogram (19). Since half the spectral intensity is necessarily lost at the half-silvered mirror of the interferometer, a Fourier transform infrared spectrometer provides only half the full (factor of N) Fellgett time advantage.] The final consideration in interpret­ ing frequency spectra obtained by Fourier transformation of a time do­ main response is ajcomparison to the spectra obtained by a conventional slow-sweep spectrometer. Under the very generally valid condition that the system response be linear (i.e., propor­ tional to the magnitude of the irra­ diating excitation), the slow-sweep and Fourier transform spectra are identical in the limit of long acquisi­ tion time. Example: Fourier Transform Ion Cyclotron Resonance Spectroscopy

Figure 8 shows the essential compo­ nents of an ion cyclotron resonance (ICR) spectrometer. The operation of this spectrometer can be understood by direct analogy to the hypothetical infrared spectrometer shown in Figure 7. A moving ion of mass, m, and charge, q, in a magnetic field, B, is constrained to circular motion at a "cyclotron" frequency, ν = qB/2itm,

[mks units; ν in Hz]

If such ions are placed between two parallel plates, as in Figure 8, and irra­ diated with a coherent (circularly po­ larized) electric field whose frequency is close to the ion cyclotron frequency, the resultant ion motion can be shown to become spatially coherent as the ions absorb energy from the irradia­ tion by increasing the radii of their cy­ clotron orbits. Once the ions are all moving essentially together, their composite cyclotron motion will in­ duce a macroscopic voltage in the sur­

Figure 11. Transient ICR signal from sample of N 2 + and CO +

rounding plates, and that voltage sig­ nal may be amplified and recorded as shown in Figure 8. The main concep­ tual difference between the ion cyclo­ tron and vibrational cases is that the system motion is circular rather than linear. For a magnetic field of 10 kG (1 tesla in mks units), ion cyclotron frequencies for singly charged ions of mass 16 to mass 400 fall between about 35 kHz and 1 MHz, as indicated at the bottom of Figure 8. It is now apparent that an ICR spectrometer provides a signal when­ ever ions of a given mass have an ion cyclotron frequency which matches that of the irradiation. In other words, the device can function as a mass spectrometer to detect ions over a range of charge-to-mass ratios, by ir­ radiating the ions with an oscillating electric field whose frequency is scanned over the range required by: ν = qB/2wm. Figure 11 shows the ion cyclotron resonance (ICR) time domain re­ sponse following excitation of a band­ width sufficient to excite ion cyclotron resonance for both A^2+ and CO + . Since the masses (and thus the cyclo­ tron frequencies) of N2 + and CO + dif­ fer slightly, the time domain response of Figure 11 is a superposition of two decaying sine waves, whose beat pat­ tern is evident in the figure. Fourier transformation of the time domain data of Figure 11 yields the ICR fre­ quency spectrum shown in Figure 12. Based on these and other (20-23) pro­ totype experiments, Fourier tech­ niques promise to reduce by a factor of 1,000 the time required to obtain an ICR mass spectrum. Conclusions

The value of any instrumental im­ provement must be gauged on the basis of its impact in making possible new experiments for experts and bet­ ter routine measurements for nonex­ perts. On this basis, Fourier methods

502 A · ANALYTICAL CHEMISTRY, VOL. 47, NO, 4, APRIL 1975

Figure 12. ICR mass spectrum obtained by Fourier transformation of transient ICR signal of Figure 11. Data for this spectrum obtained in 205 msec

have revolutionized infrared and NMR spectroscopy, by making it pos­ sible to obtain spectra of very weak signals, such as infrared spectra of planets (24) and carbon-13 NMR spectra of large organic molecules (25). Before 1965 (the advent of Fouri­ er data reduction in NMR), for exam­ ple, carbon-13 NMR spectra were ob­ tainable only with great difficulty. Today (1975), most major chemistry departments use carbon-13 NMR spectra routinely in structural and ki­ netic analysis because it now requires only a few minutes (rather than sever­ al hours prior to Fourier methods) to obtain a carbon-13 NMR spectrum. Based on the substantial proven ad­ vantages of Fourier data reduction in infrared (26) and NMR (27) spectros­ copy, the recent application of Fourier techniques to electrochemical (28), microwave (12), ion cyclotron reso­ nance (20-23), dielectric (29), and solid-state NMR (30) phenomena promises to make available to practic­ ing chemists a broad new range of ex- .

"With the Grumman ADP-30SD we now process three times as many samples without increasing our costs'.'

p e r i m e n t s n o t previously accessible. Ordinarily, t h e details of e x p e r i m e n t a l m e a s u r e m e n t , a l t h o u g h crucial t o those working in t h e field, are relative­ ly u n i n t e r e s t i n g t o chemists in gener­ al. I n t h i s article, we h o p e t o h a v e shown t h a t by t a k i n g t h e t i m e t o see how a d o u b l e - p a n balance should best be used, it is possible to e n c o m p a s s a wide s p e c t r u m of spectroscopic appli­ cations of direct chemical interest. References

Marion Walker, Health Maintenance Centers, Inc., a subsidiary of the American Health Corp., New York.

"Here at Health Maintenance we've had a tremendous increase in our laboratory load. In one year we went from 300 to 1000 tri­ glyceride and cholesterol specimen tests per week. "Even without the extra load, hand-held pipetting resulted in a high percentage of errors. Many tests had to be done twice. This was intolerable, because most of our test results are required the same day with many needed in 45 minutes. "Obviously, we needed to automate our pipetting to handle the extra diluting. After we looked at other units, we chose the Grumman ADP-30SD dilutor/dispenser. "It has now been working for us for eight months. That comes to about 32,000 tests. Errors have been significantly reduced, and I haven't had to increase our costs. An equivalent yearly saving in time of over $10,000. "NowonderlthinktheADP-30SDiswell worth the money we invested'.' For information and a free trial of the ADP-30SD, write Grumman Data Systems Corporation, Instruments Systems Products, 45 Crossways Park Drive, Woodbury, New York 11797. Or call (516) 575-3888.

Grumman Data Systems

(1) D. F.Eggers, Jr., N. W. Gregory, G. D. Halsey, Jr., and B. S. Rabinovitch, "Physical Chemistry," ρ 392, Wiley, New York, NY, 1964; W. J. Moore, "Physical Chemistry," 3rd éd., p 232, PrenticeHall, Englewood Cliffs, NJ, 1962; N. Davidson, "Statistical Mechanics," ρ 283, McGraw-Hill, New York, NY, 1962. (2) L. Mertz, "Transformations in Optics," ρ 9, Wiley, New York, NY, 1965; R. J. Bell, "Introductory Fourier Transform Spectroscopy," ρ 23, Academic Press, New York, NY, 1972. (3) P. Fellgett, J. Phys. Radium, 19, 187 (1958). (4) K. Siegbahn, C. Nordling, A. Fahlman, R. Norderg, K. Hamrin, J. Hedman, G. Johansson, T. Bergmark, S. Karlson, I. Lindgren, and B. Lindberg, "ESCA: Atomic, Molecular, and Solid State Structure Studied by Means of Electron Spectroscopy," Almqvist and Wiksell, Uppsala, Sweden, 1967; K. Siegbahn, C. Nordling, G. Johansson, J. Hedman, P. F. Heden, K. Hamrin, U. Gelius, T. Bergmark, L. O. Werme, R. Manne, and Y. Baer, "ESCA: Applied to Free Mole­ cules," North Holland, Amsterdam, The Netherlands, 1969. (5) D. W. Turner, A. D. Baker, C. Baker, and C. R. Brundle, "High Resolution Molecular Photoelectron Spectroscopy," Wiley, New York, NY, 1970. (6) SSR Instruments Co., 1001 Colorado Ave., Santa Monica, CA 90404, for exam­ ple. (7) G. Forsythe and C. B. Moler, "Com­ puter Solution of Linear Algebraic Sys­ tems," Prentice-Hall, Englewood Cliffs, NJ, 1967. (8) E. D. Nelson and M. L. Fredman, J. Opt. Soc. Am., 60, 1664 (1970). (9) J. A. Decker, Jr., Anal. Chem., 44, 127A (1972). (10) J. A. Pople, W. G. Schneider, and H. J. Bernstein, "High-Resolution Nuclear Magnetic Resonance," Chap. 4, McGraw-Hill, New York, NY 1959. (11) J. D. Baldeschwieler, Science, 159, 263 (1968); J. L. Beauchamp, Ann. Rev. Phys. Chem., 22, 527 (1971); J. H. Futrell, in "Dynamic Mass Spectrometry," D. Price, Ed., Vol 2, Heyden and Son, New York, NY, 1971; J. M. S. Henis in "Ion-Molecule Reactions," J. L. Frank­ lin, Ed., Vol 2, Plenum, New York, NY, 1972; G. A. Gray, Adv. Chem. Phys., 19, 141 (1971); C. J. Drewery, G. C. Goode, and K. R. Jennings, in " M T P Interna­ tional Review of Science, Mass Spectros­ copy, Physical Chemistry," Series One, A. D. Buckingham and A. Maccoll, Eds., Vol 5, ρ 183, Butterworths, London, En­ gland, 1972; J. I. Brauman and L. K. Blair, in "Determination of Organic Structures by Physical Methods," F. C. Nachod and J. J. Zuckerman, Eds., Vol

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5, ρ 152, Academic Press, New York, NY, 1973. (12) J. C. McGurk, T. G. Schmak, and W. H. Flygare, J. Chem. Phys., 60, 4181 (1974). (13) R. G. Brewer and R. L. Shoemaker, Phys. Rev., À6, 2001 (1972); for a very recent example of the application of infrared coherent detection methods which yield the advantages mentioned above (high accuracy in the measurement of line positions and very high instrumental resolution), see M. Mumma, T. Kostiuk, S. Cohen, D. Buhl, and Ρ. C. von Thuna, Nature, 253,514(1975). (14) R. Bracewell, "The Fourier Trans­ form and Its Applications," ρ 360, McGraw-Hill, New York, NY, 1965. (15) C. E. Brion, in "MTP International Review of Science, Mass Spectroscopy, Physical Chemistry," Series One, A. D. Buckingham and A. Maccoll, Eds., Vol 5, Chap. 3, Butterworths, London, En­ gland, 1972. (16) W. H. Furry, E. M. Purcell, and J. C. Street, "Physics for Science and Engi­ neering Students," pp 498-502, McGraw-Hill, New York, NY, 1960. (17) J. C. Davis, Jr., "Advanced Physical Chemistry: Molecules, Structure, and Spectra," pp 252-54, Ronald, New York, NY, 1965; L. Pauling and Ε. Β. Wilson, Jr., "Introduction to Quantum Mechan­ ics," pp 299-301, McGraw-Hill, New York, NY, 1935. (18) J. W. Cooley and J. W. Tukey, Math. Comp., 19, 297 (1965). (19) M. J. D. Low, J. Chem. Educ, 47, A163, A255, A349, A415 (1970). (20) M. B. Comisarow and A. G. Marshall, Chem. Phys. Lett., 25, 282 (1974). (21) M. B. Comisarow and A. G. Marshall, ibid., 26, 489 (1974). (22) M. B. Comisarow and A. G. Marshall, Can. J. Chem., 52,1997 (1974). (23) M. B. Comisarow and A. G. Marshall, J. Chem. Phys., 62, 293 (1975). (24) P. Connes, Annu. Rev. Astron. Astrophys., 8, 209 (1970); J. P. Maillard, "IAU Highlights of Astronomy 1973," Contopoulos et al., Eds., Reidel, Dordrecht, Holland, 1974. (25) J. B. Stothers, "Carbon-13 NMR Spectroscopy," Academic Press, New York, NY, 1972; G. C. Levy and G. L. Nelson, "Carbon-13 Nuclear Magnetic Resonance for Organic Chemists," Wiley, New York, NY, 1972; L. F. John­ son and W. C. Jankowski, "Carbon-13 NMR Spectra," Wiley-Interscience, New York, NY, 1972. (26) G. Horlick and H. V. Malmstadt, Anal. Chem., 42, 1361 (1970); G. Horlick, ibid., 43, 61A (1971); E. G. Codding and G. Horlick, Appl. Spectrosc, 27,85 (1973). (27) R. R. Ernst, Adv. Mag. Reson., 2,1 (1968). (28) S. C. Creason, J. W. Hayes, and D. E. Smith, Electroanal. Chem. Interfacial Electrochem., 47,9 (1973); S. C. Creason and D. E. Smith, Anal. Chem., 45, 2401 (1973) and references quoted therein. (29) G. A. Brehm and W. H. Stockmayer, J. Phys. Chem., 77,1348 (1973); R. H. Cole, ibid., 78,1440 (1974). (30) A. Pines, J. J. Chang, and R. G. Grif­ fin, J. Chem. Phys., 61,1021 (1974). Work supported by grants (to A.G.M. and M.B.C.) from the National Research Council of Canada and the Committee on Research, University of British Columbia.