topic1 in chemic~linjtrument~tion
edited by
FRANKA. SETTLE. JR.
virginlaMitarylnstiiute Lexington. VA 24450
Fourier Transforms for Chemists Part II. Fourier Transforms in Chemistry and Spectroscopy L. Glasser University of the Witwatersrand, 1 Jan Smuts Avenue, Johannesburg 2001, South Africa The chemist conceives of molecular species as having dynamic propertie-rotating, flexing, and vibrating. Typically, he examines those properties by means of spectroscopic experiment; that is, by exciting the molecular system appropriately and then examining the electromagnetic radiation that is emitted or transmitted. This radiation contains contributions from each af the many oscillators of the system, and its pattern in time may he very complicated; furthermore, the fluctuations may be so rapid 1-1013 Hz in the UV-visible reeion of the spectrum) that no detector can even begin to fdluw the detail of these fluctuations. Consequently, the time pattern of aspectroseopic wave train is (usually) ignored, and the beam of radiation is separated into ita successive frequency components by a monochromator (prism, grating, frequeneyselective detector) so that an amolitude soectrum ia eenerated and recorded. It is immediately apparent that this procedure ia necessarily inefficient: on the m e hand, if there is only one detector (often the case), then most of the incident energy is thrown away as each frequency band is selected in succession for presentation to the detector; on the other hand, a many-detector system may he very expensive (this isnot necessarily true-a photographic film is a cheap many-detector system of high sensitivity and considerable resolution hut necessitates a processing delay). By contrast, Fourier transform methods provide the advantages of either enhanced sensitivity or increased speed, or both, because (as will he explained) they avoid the process of spectral separation of the incoming signal, rather collecting all frequency data simultaneously, and substituting the FT for the frequency selector; in effect, the F T is used to scan the accumulated data to resolve the time pattern into its spectrum. This method is applicable in a wide variety of circumstances, often seemingly unrelated to the spectroscopic function upon which we have so far concentrated.
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The Number of Data Channels in a Spectrometer
Figure 2 shows a multichannel spectrometer that records the whole spectrum simultaneously, there now being N detectors. From an energy and information viewpoint this is highly advantageous, hut there is an important restriction that often makes such a system nonviable for covering a whole spectrum, This is, haw many detectors does one need? If few, because only a few spectral lines are important (say, for an X-ray fluerescence spectrometer, XRFS), then the problem is not grave--detector size erations may, however, require that the system be large so that the dispersion separates each spectral line into its own detector. If many detedors are needed, then each will
The standard scanning spectrometer is described adequately hy the diagram in Fig-
~tected, ~ n a: single A ~ esample ~ ~ a~t ~a time, ~ a ~!$":,:~ scan~,i"~~,"e~d a
ning arrangement' Apart from the small fraction of time during which any particular sample of the signal is taken, the information that is contained at that dispersion angle is simply thrown away. there is a collimation system to ensure parallelism of the beam, which throws away even more energy. This prodigality is comoensated by the fact that a sinde, broadband, smali detector can he use> to cover the whole spectrum.
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(Continued o n page A262)
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Figure 1. Blocs diagram of a a,nglbchannelspecsometer.From Crllfiths. P. R.. Ed. Transform Technrqueslrr ChamcQ; Plen~m:New York. 1978. Adapleu wllh the permlsslon of the pdollsnar and lk edslor.
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WINDOW
-'.w
N DETECTORS
m e subaanco of m.8 papa was presentea at a mln -come 'of ths SA Cnemlcal Institute. So& em T~ansvaaI~ecllon,n OcloDer 1985.
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STORAGE N CHANNELS
SPECTRUM N CHANNELS
Figure 2. Block diagram of a multichannel spectrometer. From Orlfflths.P. R.. Ed. TransformTechniques in ChemisVy:Plenum: New York. 1978. Adapted wilh me permlsslon of me publisher and the edkor.
Mlnlmum Number of Channels Required for Varlous Types 01 Munfdetecior Spectrometersa
.. Type of spectroscopy
Largest usual frequency (Hz1
Typical spectral hequency (Hz)range
Widm of one line (Hz)
minimum number 01 channels
Mdssbauer ESCA
Photoelectron Electronic Vibrational Rotational '=C NMR ICR
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Antn oilfims,P. R.. Ed. Transform rechniqms In Chemidvv: Plenum: New Yon, 1978: p 43.
hare 1.1 hp cheap and small. The table ihows the mtnimwn numbers uf detection channels required to reach standard levels of resolution in various forms of spectroscopy. The problem thus becomes: we need to mimic the multichannel device, and gain its multiplex aduontage, but use only one detector. In order to do this we must discard the disoersive element. a t the exnense of g n e m i n g n rirnmhled signal, derived from ran then multiple interferences, l ~ u which t he decoded hy the FT. The Dlsperslve Element as a Delay Devlce In order to anoroach the Fourier method. .. Irr u,rxamine conwntionol sperrmaropy in a slightly now1 way. 'l'hc function of the pri,m (the dispcriiw element in n cumentional spectrometer) is to act as a eontinuous delay element in the light path, which provides different delays for different light frequencies. This differential delay introduced into different parts of the beam Leads to destructive interference for each different frequency of the transmitted radiation a t all angles, except one that is unique to each frequency. Thus, the incoming radiation is dispersed into a spectrum. If the prism is now replaced by a grating, then similar delays and interferences occur, hut now the incoming signal may be regarded as receiving a series of finite delays (rather than the continuously varying delay of the prism), and we have a less than infinite number of beams. This results in not just one well-dispersed spectrum, as with the prism, but in anumber of more-or-less overlapping spectra, that is, the sucessive orders of the grating spectrum. Figure 3 shows the relation hetween the incoming or time-domain sienals. and the resultine snectra. In " the case uf the sampled signal, the spectra are also repeated periodically m the different orders. Thus, we see that the physical dispersive element acts, by using destructive interferences, t o pick out one frequency a t a time.
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FT Spectroscopy The Fourier transform technique takes this principle of delayed beams to its extreme hv reducine the number of beams still further; in the rase of FT-IR spectrometers, a twwheam interfenmeter is used (Fig. 4).
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Figure 3. Time domain spectral oomaln;(a) simple period c waveforms:(01 comp ex perlod c wavslwms. From GI It Ins. P. R Eo. Tmnslwm Techniqubs in Chem,shy; Plenum: New York. 1978. Adapted dth the PermlSsiOn of the publisher and the editor. Now, we no longer have angular dispersion to separate the spectral components from one another, since all beams enter and leave the interferometer in parallel streams. Instead, the relative path lengths of the two beams are altered and the component frequencies interfere a t different positions of the moving mirror, starting from the reference oosition when the two beam nath lengths are equal and all frequencies are simultaneously in phase (Fig. 5).
An analogous situation occurs when a bell is struck: all the frequencies that constitute the complex motion of oscillation are excited simultaneously, and they interfere with each other so that the sound pattern alters rapidly with time (in addition, frictional forces gradually dissipate the energy). Ifthe "bell" consists of two slightly de-tuned tuning forks then the interference will he heard more positively as a series of beats, resulting from successive destructive and construc-
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a SPECTRUM
INTERFEROGRAM
Figure 4. The Michelsan interteromster, as used for FT-IR. From Griffiths, P. R., Ed. Chemicallnfm red Fourier Transform Spectroscopy; Wiley: New York, 1975. Adapted wilh the permissioh of the publisher and the editor.
tive interferences. The NMR-FID (free-induction decay; Fig. 6)is not quite as simple as the pattern generated by a group of oscillators; it is complicated by the fact that the oscillntrons are centered around an exciting frequcncv (rather than zero). 'I'hk further scramble* the pattern hv introducing additional beats between the oscillator outputs and the slightly different generator frequency but does not alter the general relationshins. Once the interferagram or free-induction decay has been recorded, it is then possible to use the F T to decode this sienal - into its component waves. The NMR signal has interference built naturally into it; the function of the spectrometer is to excite all oscillators simultaneously and to record the "beat" pattern that they generate. In the case of the IR signal, the beat pattern is inherent in the emitted radiation, but the frequencies arb too high for our detectors; the interferometer converts the sienal from the time-deoendent pattern to sipace-dependent paitern that can mare readily he detected. ~~~~
FREQUENCY
RETARDATION
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T h e FT Advantages We have already seen that the Fourier method implies a multiplexing of the speetral information. This is called the Fellgett or multiplex advantage. It is, however, only an advantage when time averaging of the signal leads to enhancement of the signalto-noise ratio. This advantage occurs when the noise strength is constant, independent of the signal strength; such noise originates in the detector when the photon energy is less than kT, and is called "detector-limited" noise. When the photon energy is large (e.g., in the UV-visible region), however, the detector introduces essentially no noise, and the noise originates from the source, so that
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Figure 5. Fourler Transform infrared iFT-IR) . . soectra:. la) . . simole soectra:. lb), corresoondina" interteroorams: " . (c)a complex imerteragram. From Griffiths. P. R.. Ed. Chemical bfraredFourier mnsform Spectroscopy; Wiley: New York. 1975. Adapted with the permission of the publisher and the editor.
I I
TIME
Noise = (Signal)"2 Such noise is termed "source-lithited" and there is no multiplex advantage to be gained a t all. If the noise strength is proportional to the signal strength ("fluctuation noise"), there is actuallv a disadvantaee in wine" multiplex procedures. F o r t u n a t ~ l y , ~ h e n , t ~ l tiplrx ad\nntage dues upcrate for the weak JcrllrCQS characteristic of infrared and N M R Figure 6. Fourier Transform Nuclear Magnetic Resonance (FT-NMR) spectra: (a) tree-induction decay (FID); spectroseopies. (b) corresponding spectrum. From Griffiths, P. R., Ed. Transform Techniques in Chemistry: Plenum: New (Continued on page A264) York. 1978. Adapted with the permission of the publisher and the editor. Volume 64
Number 11
November 1967
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Another advantage of Fourier spectroscopy with a Michelson interferometer o m hecause of the absence of an entrance slit, for -~~there is no need to limit the beam width in . .. order ..-. to .. nhtain adeauate resolution (although there are rertain, lessstringent, limits to beam width). Aa a consequence, there is a "throughput" or Jacquinot advantage that corresponds to an enhancement of the beam intensity. It is no wonder that the first use of F T spectroscopy arose in circumstances of very low signal strength, such as in stellar spectroscopy or 13CNMR. ~~~~
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Computing the FT: The FFT A serious problem in early applications of FT was the labor of calculation, and this grows rapidly with the number of data points; thus,Ndatapoints require W multiplications (all N data points contribute to each of the N transformation points). This calculation became practical with the advent of computers in the 1960's, but the nroblem was still not insimificant because ihe typical number of data points is not small. aav 20(KI or B O W in a ervstal structure determination and 10,000 or even 20,000 in an FT-IR or NMR experiment. Fortunately,Cooley and Tukey (1965)developed an algorithm for performing the F T in a most ecoiomical manner, relying on the opportunity to factor the data into sparse matrices. that is. contninine mosth zeros. Not onl; is this &onomical in time;requiring only N log2 N cumplea multiplications (Fig. 7)-which is a gain of NAogz N-but it ~
cess simply reduces to a sequence of shuff l i i of data points and recombining them in pairs. The FFT is not entirely without restrictions, which need to be considered when deciding on its use. Perhaps the most critical feature is that the FFT (in common with its parent, the digital FT-DFT) is not an exact replica of the FT, in that the sampled and digitized FT treats the data set, in effect, as part of a periodically repeated signal. This has no serious consequences if the original signal is limited in extent, for example, if its snectrum cuts off to zero a t accessible freakncies. and also nrovided that the sienal " is sampled sufficiently frequently, that is, at least twice per cycleofthe highest frequenry (thisis termed the Nyquist criterion). lithe signal is undersampled, then the higher frequencies will fold over into the lower frequency region and mimic, or "alias", lower frequency terms (Fig. 8).In practice, no signal is truly hand-limited; this has the consequence of aliasing, but sufficient care in handling the data (e.g., filtering out higher frequencies hefore sampling) will generally reduce aliasing to manageable proportions. A second restriction on the FFT is that it requires data to be coileeted a t even intervals. With eomputer-controlled sampling, this is generally acceptable, but it is, of course, a restriction on its use. Finally, the FFT ismost efficient and easily handled when there are Zrn data points (m = integer), such as, 1024,2048, etc. This is actually quite easily overcome simply by filling the data array with zeros ta complete the set of 2"' values. Such 'kero-filling" is advantaeeous because it has the effek of interpolating in the transform domain. Thin advantage arises because of the nature of the FFT; each (cmnplea) data pomt in the original set gives rise to a (complex) data point in the transform domain, with resolution determined by the length of the data set in the original domain--increasing the length of the one interpolates in the other. Of course. this cannot be carried on indefinitely and the ultimate resolution is determined by the highest frequency component sampled in the original signal. The FFT is not universally useful. Since it is discrete, it yields a transform a t predetermined discrete intervals only; if these intervals are unsuitable (e.g., to plot out a function in detail) and if onlv a relativelv few vwsform points are required, then it may be preferable ta generate the transform by the brute-force or "slow" FT. ~~
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Figure 7. m e Fast Fourier Transfwm (FFT) advantage: (a) number of mmplex operations In a Oiamete Fowler Transfoim (DFT)(=N2)and In an FFT (=N log. N): (b) relative number (=N/lcg2 N ) of complex operations in DFT as against FFT. can also be done in place, requiring no further data storage space than the space required for the original data set. The FT pro-
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Some Applications In Chemistry The applications of F T in chemistry are manifold and it seems, on examining the literature, that there are few techniques to which FT's have not been introduced at one stage or another, either in the data collection process (which is our present concern) or in data manipulation (which is the subect of our ensuing discussion). One of the classical uses of FT is in X-ray crystallography, where the technique was introduced in some of the earliest crystal structure analyses. It is somewhat odd that the standard references on F T in chemistry barely mention, if they do a t all, X-ray and electron diffraction usaee: nerhans . this is because diffraction is so thoroughly covered by its u r n literature. ~~~~~
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Figure 8. Aliasing; (aja serial domain signal of frequency r sampledat the Nyquist frequency, that is, Wlce per cycle; (b) the samples, at the same lntewal on a smat of frequency s+ A%(o)the samples on a signal of hequency s - Asare identicalto those on the signal of frequency s As; (d) hequency domain spectral commmnls "folded back" to lower freouencies as the carrier freauencv Is oraduallv increased (from bonomlolop181canstanl sampling frequency Ihra~ghtheNyqust hequency-in thiscase, accompan ed by a cnange of phase From Griff;tns.P. R.. Ed. Trsnstonn Technlqueo b Chemishy; Plenum. hew Vorr. 1978. Adapted with me permissionof the publisher and the edha.
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From the point of view of F T procedures, diffraction processes (whether optical, Xray, neutron, or electron) can he regarded as incomplete image producing procedures, which are later completed by the FT. In common with other optical processes, a beam of radiation impinges on a sample and is diffracted by it. This diffraction pattern is the Fraunhbfer or Fresnel pattern (an optical transform), and corresponds approximately to the FT of the object. This can he seen most readily ifwe consider the simplest diffraction process as involving a monochromatic beam of radiation impinging on a grating. The diffraction "spots" that the grating produces are images of the source of radiation, with angular displacement inversely related to the grating spacing, that is, the smaller the spacing, the wider the angle of diffraction (Fig. 9). A grating of complicated three-dimensional structure (a crystal bathed in X-rays) will give rise to a complicated diffraction pattern in the whole of angular space.
Figure 9. Effects of spacing in the object an a diffraction pattern; lefi-slits; right-diffraction panern. From Taylor. C. A. Images; Wykeham: London. 1978. Printed with the permission of the publisher and the author.
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Number 11 November 1987
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The function of a lens is to collect the diffracted radiation and perform a further FT to create an image, which may he of different size to the original ohject. In the case of X-ray diffraction, no lens exists (since a lens requires a refractive index, n, greater than that of air, while the refractive index of solid materials is close to unity), so that the lens must he substituted by a process of collecting the diffraction patternphotographically or by diffractometer-and then subjecting the collected data ta an FT. Unfortunately, all possible detectors record the diffraction intensity hut lose the phase information of one "spot" relative to another (more will be said of this matter later), so that the automatic reconstruction that a lens performs is not available in the mathematical FT process. Instead, the phase information must he supplied either by inference, using the Patterson function as a guide, or by statistical procedures, as in present-day "direct methods". An interesting use of direct recording of a diffraction pattern is holography, where the interference pattern hetween a diffraction pattern and a reference beam is recorded photographically. The recorded hologram is now a complex grating, which can be used to reconstruct the set of beams that created it, and so provide a three-dimensional image of the original diffracting ohject. Optical transforms have also been used to mimic the mathematical FT process in the reconstrue-
t i m i,f images from electron diffraction pmresressnd t o correct ur allow for problemaof focus in electron microscopy. FT techniques were first introduced, in the more generally recognized chemical sense, in the 1950's when Connes and Mentz develooed both hieh-resolution and raoid" scanning, law-resolution infrared interferometers for optical astronomy. Gebbie and Strong pioneered in work in the far infrared. The FT techniques in these optical fields rely an interferometers of various kinds. In NMR spectroscopy, the appropriate FT teehniques are based on pulse methods; these were pioneered in the late 1940's with the availahilitv of radar eouioment and correspunding expertiie tu promote the proredures. I t is interesting that surh puke techniques are by no means as new as might he thought. The original procedure for measuring a capacitance is to apply a step voltage and determine the time constant (IIRC) of the decay of the voltage to its final value; this clearly involves a transform from the time domain to the frequency domain, where the time constant is really a reciprocal rate constant. I t is clearly not appropriate here to attempt to describe in detail how specific F T procedures are performed and their analysis, so we will omit further consideration of the procedures of FT-IR and -NMR. Rather, it is preferable that we spenda little more time in considering other applications of FT methods in chemistry. One field of particular interest in this context is electrochemistry, which has been characterized1 as peculiarly suited to FT treatment since:
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(1) direct digitization is possible, because of the suitable frequency range involved (dc to MHz); (2) test signals and input waveforms can readily he tailored to needs; (3) electrochemical response spectra are weakly structured, so that there is no need for high spectral resolution (in contrast to the needs in standard forms of spectroscopy); and (4) the nature of the electrochemical response yields rewarding applications, for examole..in studies of kinetics and mechanisms, transient behavior, and double.layrr properties. ~~
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Other fields in which FT's have successfully been applied include, in general, techniques in which some pulsed signal can he applied to generate interference (''heat',) resoonses. Examoles include the followine types of speclrmcopie~:ion cyclotnm resonance ra form of massspertrom8py). nurlear quadrupole resonance, dielectric response, microwave response, electron spin resonance, and muon spin rotation. Of course, the chemical applications by no means exhaust the field and there is an enormous numher of applications in the fields of physics (e.g., diffusion), electrical engineering (e.g., antennae), statistics, image processing and enhancement, and so on. It will be the function of the next section of this study of the FT to show how certain useful properties of the F T make it so generally applicable. 'Smith, D. E. Anal. Chem. 1976, 48, 221A, 517A.
