Alan G. Marshall and Melvin B. Comisarow University of British Columbia Vancouver. Canada v6T 1ws
Fourier Transform Methods in Spectroscopy
A multi-channel spectrometer is a desirable instrument from the viewpoint o i signal-to-noise considerations. While direct multi-channel spectroscopy is impractical (except for optical and photoelectron spectra), a sihematic analysis of the Fourier transform technique will show that this encoding ("multplex") method effectively opens up the slit width of a spectrometer to embrace the full spectral range a t once without sacrificing resolution. In this paper illustrative spectral lineshapes are shown, and an experimental example is drawn from Fourier transform ion cyclotron resonance spectroscopy, the most recent application of transform techniques in spectroscopy. Signal and Noise In any experimental measurement characterized by a certain level of imprecision or random noise, it is desirable to repeat the measurement many times in order to obtain a more accurate result. The signal (in this case, the intensity of a spectroscopic absorption "peak") will accumulate as the number of spectral scans, N; but if the noise a t any particular point (frequency) in the spectrum is random, then the noise amplitude a t that frequency may he treated simply as a random walk about zero (the auerage noise level), and it is well-known that the average absolute distance away from zero after N steps of a random walk (more precisely, the root-mean-square distance) is proportional to 477 ( I ) . Thus the true measure of precision of the repeated measurement, the signal-to-noise ratio, is proportional In this paper, w will present the to ( N l m , or just 4'. direct (''multi-channel") and indirect {Fourier, "multiplex") methods for attaining the full d N improvement in sienal-to-noise ratio available in various types of spectroscgpy, and some discussion of the significanEe of this recent improvement, Direct Multichannel Soectrometers Figure 1 is a highly schematic diagram 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 that when detector readings are collected from a number of individual slit positions (bottom of Fig.
Figure 1. Top: Schematic diagram of a single-slit scanning sbsorption spectrometer. A singie-slit scanning emission spectrometer lecks only the broad-band source. Eonom: Detector readings hom a number of individual slit poshions.
638 / Journal of Chemical Education
bmad debdm N
bond
INchamPhl STCRAGE
F l g m 2 Scnamatoc oagram of a dorect m.tchannei spectrometer, pored of many reparale nngle-channel detectors
cam-
I), there is sufficient resolution to distinguish spectral features of interest. The most important feature of such a spectrometer is that its detection of an absorption spectrum requires a procedure in which only one possible slit nosition (i.e.. one narrow s ~ e c t r a window) l is ooen at anv given stage df the spectral scan. I t would thus de desirabie to onen un the slit aoerture to the full width of the desired spectral window, by using N separate single-channel detectors as shown in Fieure 2. Since all slit wositions are now monitored a t once, rather than just one a t a time, the spectrometer of Fieure 2 offers (in princiwle) an improvement of the full v'%' advantage in kignal&-noise ratio, compared to the result of a single complete spectral scan requiring the same total time by the spectrometer of Figure 1. Alternatively, it would be possible to obtain a spectrum having the same signal-to-noise ratio in (l/N), the time required to scan the N individual slit positions one at a time. Because of the conceptual simplicity of the spectrometer of Figure 2, i t is logical to investigate its feasibility. It is desirable to be able to resolve spectral detail as narrow as the width of a typical spectral absorption line; therefore, the minimum number of channels that will be required in a multichannel spectrometer is simply the width of the entire spectral range of interest, divided by the width of a single spectral line. The resultant necessary number of channels for various forms of spectroscopy is shown in the table. From the table. i t would aooear that electronic (visibleUV)spectroscopy 'is the least' iikely candidate for 'success with a direct multichannel spectrometer; but in fact, multichannel detection of optical-uv radiation is readily accomplished ohotoera~hicallv.The resolution of a fine-erain photographic plate is sufficient to provide for the i u g e 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) (2,3) and whotoelectron spectroscopy (2. 3), electrons are dislodged from atoms or ~ o l e c u l e s ~ hX-ray y 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 electronic states can be determined. By passing the electrons between
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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.
two charged parallel plates, the dislodged electrons may he disnersed in mace accordine t o their velocitv. ?.to achieve t h e arrangement shown i n - ~ i p r e2. This multichannel electron-detection scheme has recentlv become feasible with the advent of the vidicon detector,i in which an arriving electron strikes a fluorescent screen on the face of a tefevision camera. Since electrons of different velocity can he disnersed to strike different regions of the screen, their arrivai will he recorded independently by different elements of the television camera grid. Because of the small required number of detector channels (see the table), the Figure 2 spectrometer is thus now feasible for ESCA and photoelec&on spectroscopy. With the other forms of spectroscopy listed in the tahle, direct multichannel methods are less attractive. For microwave (rooltional) spectroscopy, for example, there is no hroad-hand radiation source available: A black-bodv radiation source, such as employed for other radiation energies (xenon or hydrogen discharge for uv, hot tungsten wire for visible, glohar for near infrared and infrared, mercury vaDor for far infrared) would have to be operated at an unreasonably high temperature in order to-obtain sufficient radiation flux for use as a radiation source. I t would be conceivable to construct an array of individual (narrow-band) microwave transmitters (about $5,000 each) as the "broadband" radiation source, but the tahle shows that the cost would be excessive ($lo9 . . . !). For infrared spectroscopy, on the other hand, the necessary broad-hand source is available, hut i t would he necessary to disperse the spectrum over many meters in order to he able to resolve the desired spectral detail with existing individual (thermopile) detectors of about 1 mm width each, and at a cost of ahout $200 per detector; the total cost again hecomes unmanageable. (Photographic detection does not extend beyond about IZ,OOOA,and is thus unavailable.) Finally, for nuclear magnetic resonance (nmr) and ion cyclotron resonance (icr) spectroscopy, broad-hand sources are again available, hut the cost of an array of tens of thousands of individual narrow-hand mixer-filter detectors (see below) is again unreasonably high. For infrared. microwave. and radiofreauencv. mectrome. ters, then, the dirrcr multichannel approach is just not feasihle. either eeometricallv or financiallv. We will now consider a recent and valuable indirect approach: Fourier transform spectroscopy.
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Fourier Transform Spectroscopy
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 frequency domain-we judge light by its color and sound by its pitch. I t is, however, equally useful to analyze observations in the time-
Figure 4. Frequency representations (right) of three types of limedomain (left) spechometer signals-see text
M l n ~ m u mNumber of Channels Requ~rsdfar Varlour Types of Dlrect Muitmhannei Specttometerr
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--
Largest Usuai T y p e of Frequency S P ~ C ~ ~ O S C O P Y (Hz)
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~ypicsl Spectral Range ( H z )
Width of One Line (Hz)
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Minimum Number of Channels
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~b;srbauer ESCB
Photoelectron Electronic Vibrational Rotational "C N m r 1cr Number o f channel5 is obtained b y dividing the typical roectral range b y the width of one spectral hne.
domain, as shown in Figure 3. The upper-left diagram of Figure 3 shows a simple dc (zero-frequency) pulse, which is turned on a t time zero. and turned off at time. T.Intuition would suggest that the frequency representation of such a pulse should consist simply of a signal which is spread over a range of frequencies near zero. By using a shorter pulse (middle of Fig. 3), the frequency representation is spread over an even wider range, and in the limit that the dc pulse is made infinitely narrow (bottom of Fie. - 3). . the freaueocv representation is-a completely flat ~ p e c t r u m . ~ The diagrams in Figure 3 suggest that the broad-band frequency excitation required f g a multichannel or multiplex spectrometer can he generated by use of a sufficiently narrow electromagnetic radiation pulse. (If the pulse coosists of an ac, rather than a dc waveform, then the pictures of Figure 3 still apply, except that the frequency representation is now centered a t the ac frequency, rather than a t zero frequency-see top trace of Fig. 4.) For nmr, for example, the table indicates that an excitation bandwidth of trace) indicates ahout 10 kHz is rewired-Fieure 4 (ton . that such an excitation may he produced simply by applying a radiofrequency pulse whose duration is of the order of 10 fis. As another example, electron impact spectroscopy (6) is based on the rapid nassaee of an electron Dast a molecule-this passing eiection p;oduces a very short, sharp pulse of electric 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
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Figure 3. Time-Domain (len) and frequencydomain (right) representations of de pulses of three different durations.
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'SSR Instruments Co., 1001 Colorado Avenue, Santa Monica, Calif. 90404, for example. 2The mathematical correspondence between the time-domain and the frequency-domain diagrams of Figures 3 and 4 is called a (one-sided)Fourier transformation (5).
Volume 52. Number 10, October 1975 / 639
same sort of transitions as are more conventionallv studied in photoelectron and ESCA spectroscopy. When a given single oscillator is subiected to irradiation at its resoiant freq"ency, the amplitude of oscillation will increase. If the irradiatina excitation is then removed. the oscillation will persist with an amplitude which decreases (usually exponentially) with time, as shown for three convenient limiting situations a t the left of Figure 4. If the oscillation is not appreciably reduced during the acquisition time, T (top trace of Fig. 4), then the corresponding frequency representation has a functional form which resembles the amplitude of (Fraunhofer) diffraction by a slit (7). If on the other hand the oscillation is observed for seueral lifetimes of its decay (middle trace of Fig. 4), the spectral representation approaches the familiar Lorentzian line shape encountered in manv forms of s~ectroscoov.Finallv. the-bottom trace of ~ i ~ u 4r illustraks e the i&rmediaid case, in which the acquistion time, T.is of the order of the decay lifetime, r. he-irreversible deeay of the oscillation is due to: radiative damping ("spontaneous emission") (8); and to interactions of the sample (nucleus, ion, molecule) with its surroundings, where the interaction may he neutral-neutral collisions (microwave, infrared, optical); ionmolecule collisions (ion cyclotron resonance); rotational diffusion (nuclear magnetic resonance, electron spin resonance); or depletion of the excited species due to chemical reaction. The multichannel advantage of the Fourier approach can now be understwd. Suppose that the time-domain response, yft), is sampled a t N equally spaced intervals during a total acquisition time. T.Each of these samoled timedomain points, y(tJ, n = 1to N, is then a lineaicombination of all the discrete frequency-domain spectral points, +(urn), according to
a,,
=
exp [Zrim tJT1
(2)
a,,,
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exp [2nimn/N1
(3)
or just
Since there are N known independent observed sampled time-domain amplitudes, y(tJ to y ( t d , each expressed in terms of all N desired discrete frequency-domain spectral intensities, z(u J to ~ ( w N ) , it is possible to "decode" the observed data to obtain the desired spectrum (simply by solving the N linear homogeneous algebraic equations in N unknowns of eqns. (1)). For the particular coefficients, a, of eqn. (2), this decoding procedure is called a discrete Fourier transformation, and may he calculated rapidly hy a digital computer (9). Finally, since the magnitude of each a,, in the Fourier experiment of eqns. (1)is unity
it is as if each spectral intensity component, x(wd, is being detected with unit weight factor. In other words, in the Fourier experiment, it is as if all the possible spectrometer slits are open at once, with the sonrr spectral resolution that would have been obtained hy opening just a single slit! Hecause all N possible "channels" or "slits" are open at the same time, it is now clear by the argument given a t the very beginning of this paper that detection of the time-domain response, followed by Fourier transformation ("decoding") to obtain the frequency-domain spectrum, provides a fre640 / Journal of Chemical Education
Figure 5. Ion cyclotron resonance (icr)mass spectrum obtained by Fourier transformation of a transient icr timedomain signal from an approximately equimoiar gaseous mixture of N2 and CO. The data for lhis spectrum were obtained in 205 ms. More recent experiments have produced resdutions of 70,000 1251 end 250,000 1261. quency spectrum exhibiting either (a) signal-to-noise improvement of a factor of flin the same total observation period, or (b) a spectrum having the same signal-to-noise ratio in a factor of (UN) as much time as required by a conventional spectrometer which scans the spectrum slowly with a narrow-bandwidth d e t e c t ~ r . ~ . ~ Example: Fourler Transform Ion Cyclotron Resonance Spectroscopy Ion cyclotron resonance (icr) spectroscopy is a type of mass spectroscopy with particular value in the study of ionmolecule reactions in the gas phase (12). The feasibility of Fourier transform icr spectroscopy has recently been demonstrated (13-16). Figure 5 shows an icr mass spectrum obtained by Fourier transformation of a sampled time-domain icr response for an approximately equimolar mixture of Nz and CO. Based on this and other prototype experiments (13-16), Fourier techniques promise to reduce by a factor of 1,000 the time required to ohtain an icr mass spectrum. Conclusions The value of any instrumental improvement must be gauged on the basis of its impact in making possible new experiments for experts and better routine measurements for non-experts. On this basis, Fourier methods have revo3The signal-to-noise ratio for the decoded spectrum will he hetter than that from a single-slit scanning spectrometer, only if the noise at any given frequency is independent of the signal at that frequency. This situation is called "detector-limited" noise, and is distinguished from "source-limited" noise in which the root-meana l the sauare root of the sienal maenisauare noise is ~ r o. ~ o r t i o n to tude. When noire la sourre-lrmited, the above encoding.decodinl: method will nor improve the signal-to-nomeratio of t h e decoded spectrum over that determined in the conventional one-slit-nt-atime scanning procedure (10). Spectroscopic examples in which the noise is detector-limited include: infrared, microwave, nuclear magnetic resonance, and the ion cyclotron resonance experiments, while examples in which the noise is source-limited include: optical (visible-ultraviolet) and chareed-oarticle (ohotoelectron. ESCA. " . electron impact) spectroscopy. 'For the unique case of Fourier-transform infrared speetrometers based on the Michaelson interferometer, the spectrum is obtained by discrete Fourier transformation of the (spatially disnersed) samoled interferoeram (11). Since half the soectral intensity is ne~essarilylost at the half.silvcred mirror uf the intrrfcrom. rter. a Fourier transform inirnred spectrometer provides only half the full ifaetur of N I Fellgetr time advantage.
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lutionized infrared and nmr spectroscopy, by making it possible to obtain spectra of very weak signals, such as in-. frared spectra of planets (17) and carbon-13 nmr spectra of large organic molecules (18). Before 1965 (the advent of Fourier data reduction in nrnr), for example, carhon-13 nmr spectra were obtainable only with great difficultytoday (1975). most major chemistry departments use carbon-13 nmr spectra routinely in structural and kinetic analysis, because it now requires only a few minutes (rather than several hours prior to Fourier methods) to obtain a carbon-13 nmr spectrum. Based on the (substantial) proven advantages of Fourier data reduction in infrared (19) and nmr (20) spectroscopy, the recent application of Fourier techniques to electrochemical (21), microwave (22), ion cyclotron resonance (1>16), dielectric (23), and solid-state nmr (24) phenomena promises to make available to practicing chemists a broad new range of experiments not previously accessible. Ordinarily, the details of experimental measurement, while crucial to those working in the field, are relatively uninteresting to chemists in general. In this article we hope to have shown that by taking the time to see how Fourier transform methods work, it is possible to encompass a wide spectrum of spectroscopic applications of direct chemical interest. Literature Cited
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u' Hahey. Jr . C D..and Hahmoviuh.B s ."Phy.L. ~ aCl h e m , ~ ~ q , " J o hUt1.y n & +IN. Neu Y x k . 1 % & p 392 M u m . W J .'Physical Chrmmfr)." : t d Ed. IPnnt~r.Ha.1.Englcavvd Cltiir. N J.. 1962. p. 232. I h " Hcx.k Company. Yea Y o r k . \~dcnn 1.''Srsu.t~~rlM e r h s n , ~ ~ .M&rsw.H.II ,962. p LPC 21 S w l r h n . K. 1ordhng.C. Fahlman. A . Nwdem. R.. Hamnn. K , Hedmsn, J. Jo. hsnason. G R s m s r k . I'. Karlron. S.. l . m d p n . I . and Lindkrg. H 'ESCA lo m r . Molcrul8r,and Solid S t e v S r r u n u r c i t d i e d b, m r a n 8 d E I ~ l m n S p r c , U p w l a . 19R7. tru~,~.py,"Alrnqu~rt m d \V~l;cllr U o k t r y c k ~ rAB. ,:I, s,e#hahn. K . Nordllng C .J,.brnmm (i klrdmrr,. J.. Hedcn. I' F . H 0 , i n . K.. I , F.gols. J , . 11. F . G,e**ry. N.
.
.
Gelim,U., Bergmark, T.. Weme, L. 0.. Manne, R.. and Bapr. Y., "ESCA: Applied to F m Malemle8." North Holland Publiahiq Co., Amstordam. 1969. (4) Twner, D. W., Baker. A. D., Baker, C., and Brundlc, C. R.."High Resolution Molmular Photcdeetron SpeBoeeopy." John Wilcy& Sons. Nnw York. 1970. (5) BrsawieU. R., '"The Fourier Tranafarm and 1- Applieationa." MeGraw-Hill Bmk Co., NewYork, 1966,~.360. (6) Brim. C. E., in "MTP International Review or Seicnee. Mass Speetroeeopy, Physi4 Chemi*rv:'Serie~ One. Vol. 5. (Editors: Buckinehem.A. D.. and Maemll. A.),
York. 1935, pp. 299-301. Cooley, J. W..and Tukey, J. W., Moth. Comp. 19,297 (19651. 8 ~ ~L..~"Tmnsrormations ~ h , in Optics." John Wiley & Sons. New York. 1965, p. 9: Bell, R. J., "lntraduetory Fourier Trsnaform Speetroneopy: Acsdemb Prass. New York, 1WZ.p. 23. I Low,M. J. D., J. CHEM. EDUC.. 47.A163, A255, A349, and A415 (19701. I Braurnan. J. I.. and 61air, L. K., in "Determination of Organic Structures by Physi'nl S (Editor: Nachod. F. c.. and Zuckorman. J. J.). Acsdemic
. ~.
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.
(16) Comiiarow. M. B.,and Marshal1.A. G.. J. Chem. Phyx.. 62.293 (19751. (17) Connes, P., A m . R.u. Asfmn. Aatrophys., 8.209 (1970): Maillard, J. P.."IAU Highlights of Astronomy 1973." (Editors: Contopoulos, et d.1. D. Reidel, Oodrmhf, 1974. (18) stothem. J. B., "carbon-13 NMR spctmswpy." ~ ~ ~press. d N.Y.. ~ 1972: ~ Levy. i l G. C., and Nelson, G. I., '"Carbon-13 Nuelear Magnetic Resonance for Organic chemists." John Wiky and Sons, NewYork, 1972;Johnson, L. F., and Jankourki, W. C.;~Carhon-I3 NMRSrntra." Wiley-Interscience. New York, 1972. (19) Halick G.,and Malmatadt, H. V., Anol. Cham.. 42,1361 (19701:Horlick,G.. Aml. chon;., 13,61A (1971): Cdding, E. G., and Hodick, G., Appl SpecfrOSe., 27.85 (1973). (20) Ernst. R R.Adu. Mag. Rpa., 2.1(19681. (21) Creamn. S. C., Hayes, J. W., and Smith. D. E., Elec~momlyf.Chem. and 1nferf~cia1 Electrochem... 17.9 . (1973): Crranon. S. C.. and Smith, D. E., Anol. Chem., 45, 2401 (19731,and referanan therein. (22) M&urk, J. C., Sehmalz. T. G.. and Rygare. W. H . , J Chom Phw., 60.4181 (1974). (23) Brehm, G. A.,andStaekmayer, W. H., J. P h y s Chem.. 77, 1348 (1973): Cole, R. H., J. P h s . Chom.. 78.1440 (1970. (241 Pines, A,, Chang, J. J., and Griffin, R.G., J Chem Phys., 61,1021 (1970. (25) Manhal1.A. G., and Comisarow,M. B.,Anol. Chsm., 47.491A 119751. s lG., l 23rd . Annual Conforenee on Mars Spectres(26) Cominarow. M. B., and ~ ~ ~ h A. ,py and Allied Topia, arranged by the American Society for Mass Speetrometry, Houston.Teras. May 1975. p s p r R-5.
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