fourier transform-infrared spectroscopy - ulisboaweb.ist.utl.pt/berberan/pqf/jce perkins...

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FRANK A. SETTLE, JR. chemical in/trument~tion Lexington. VA institute 24450

Fourier Transform-Infrared Spectroscopy Part I. Instrumentation

W. D. Perkins The Perkin-Elmer Corporation, 41 1 Clyde Avenue, Mountain View, CA 94043

The foundations of modern Fourier Transform-Infrared Spectroscopy (FT-IR) were laid in the latter part of the 19th centu- ry. The renowned phsyieist A. A. Michelson constructed an interferometer and described it in publications in 1891 and 1892 ( I , 2 ) , and as early as 1892 Lord Rayleigh (3) recognized that the interferogram was related to the spectrum by a mathematical oper- ation known as a Fourier Transformation. The technique was horn before its time and was little used for more than five decades while awaiting the development of digital computers that could perform the transformation rapidly and economically.

Peter Fellgett, in 1949, is credited with heine the first to transform an interfero- ~~~~ n~~ ~ ~~

gram into i t , rorrrsponding infrared spectrum ( 4 ) . Rut computers in the 'f~fties were slow compared to today's models, and com- puting times ranged from minutes into hours. Furthermore, computers then were too expensive to be dedicated to a single laboratory instrument and remained in eomouter centers. The soee t rasc~~is t had to ~ ~

bring his intderogram recorded on tapr to the computer center and leave it there for procesrrng. Often i t was not until che follow ing day that he saw his spectrum.

A tnakthrough came in 1965 when two mathematicians named Cooley and Tukey

W. D. Perklns received his B.S. degree in chemistry hom the Massachusetts institute of Techmlogy in 1948 and pursued graduate stwlies at Harvard University where he received an M.A. in physics in 1950 and a PhO in chemical physics In 1952. Following several years as a re- search technologist in the Shell Oil Re- search Laboratory at Martinez. CA, he joined the inhared product department of The Perkin-Elmer Corporation in Norwalk, CT where he was active in instrumentation haining courses as well as in product development. evaluation. and testing. In 1982 he was transferred to Perkin-Elmer's Mountain View. CA. office where he cur- rently holds the position of Senior Prcduct Specialist. Infrared. Since 1975 he has also been on the staff of the Applied Mc- lecular spectroscopy Program at the Ari- zona State University in Tempe, AZ. His Current interests involve the application of FT-IR spectroscopy to a variety of industri- al problems.

FIXED POSlTiON MIRROR

MOVABLE MIRROR

d - I I I I

SiNGLE FREQUENCY SOURCE ( A ) 3

8 = A 8 . - A 2 ! !

DETECTOR 9 I I

I !

Fiaure 1. Block diaaram showina the mior comoonems of an FT-IR soectrometer. The fiaure also shows - - Me roechum of an infinitelv narmw line sourcs and hcw the intertermram is oenerated as the movable mwror istrans ated Mar mainthe nterferogramoccbr wnen Meretaroat on r equal loan integral mu tiple of tne uavelengln of the source M nlma occur when W e retaraatmn is an odd mdl pie ol nall wave sngths

develooed an aleorithm that hears their " names and which speeded up the cornputa- trun of Fourier Tranrfurrnatiun timrs bv nearly an order of magnitude 15). Its appli- carim to interferunwtrv was rrcognilcd quickly by Forman ($1. 'l'hc sragc was now set, and during thc mid-1960's a few eom- mercial instruments appeared on the market for use in the far-infraredspectralregion where mechanical tolerances for the interferometer uere lpsr $were than in the mrd. mfrnr~d and u here the rcqurrement. fur the computer were also more lenient.

By 1969, benefitting from space-related military technology, one manufacturer was able to market a low resolution, mid-infrared- (4000-400 cm-'1 range interferometer that included a fully dedicated computer and that sold for between two and three times the price of a high performance dispersive instrument. During the intervening years mechanical designs have been im- proved and computer costs have decreased dramatically while speed and memory size have increased significantly. Mid-infrared- range FT-IR's are now available in the me- dium price range with resolutions of one wa-

venumber and better and with dedicated computers ahirh can trsnshrn, a sprrtrum in a, littlcasO.2 ~ 0 0 . 3 s with memories large enough to accommodate extensive data processing programs.

Instrument Deslgn

Despite the developments in instrumentation, the basic optical design used origi- nally by Michelson is still prevalent (see Fig. 1). Energy from a conventional infrared source (a heated element or glower) is colli- mated and directed towards a beam splitter. In the mid-infrared range the beam splitter is usually a very thin film of germanium supported on a potassium bromide suh- strate. An ideal beam splitter will reflect 50% of the incident lieht and transmit the remsinine 50%. thus crestine two seoarate ~ ~ - , ~ ~~~ "~ ~~~ . ~~ ~~~~

optical paths. In one path the beam is reflected by a fixed-position mirror back to the splitter where i t is partially reflected to

(Continued on page A6)

Volume 63 Number 1 January 1986 A5

the swrreand partmllv tmnwmtted and fo- cwed onto the detertur In the other leg of the interferometer the beam is also reflected, this time by a movable mirror that can be translated back and forth but that is always maintained parallel to itself (i.e., without tilt or wobble). The beam from the movable mirror is also returned to the beam ~ ~~~

splitter where it, too, is partially transmit- ted back to the source and partinlly reflected to the detector.

The energy that reaches the detector is the sum af these two beams. If the distance from the center of the beam splitter to the fixed mirror is the same as the distance from the beam splitter to the movable mirror, then the two beams will have traveled equal distances. When the second mirror is moved, the optical pathlengths become un- equal. The path difference is known as the retardation or optical path difference (O.P.D.) and is commonly represented by the symbol 6. I t can be seen that if the movable mirror is displaeed a distance x , the retardation is 6 = 2% sinee the light has to travel an additional distance r to reach the mirror and then another distanee x in the reflected beam to reach the paint where the mirror was before being moved.

Consider the output signal from the detector when the souree is emitting a single frequency (a single frequency laser, for ex- ample). When the optical path in both legs of the interferometer is identical (6 = O), the two beams reaching the detector will be in phase and will reinforee each other. The intensity of the detector signal, represented bythe symholI(6) (see table), will be amaxi- mum.

Abbrevlatlons and Termlnology

FT-IR

X

6 ow ZPD l(6)

B(4

FFT

centerburst

DTGS

MCT

A

FWHH

ILS

Fourier Transform-lnfrared-aI- ways hyphenated

Displacement of movable mirror Optical retardation (6 = 2x) Optical Path Difference Zero Path Difference Intensity of the interterogram as a

function of 6 intensity of the spectrum as a func-

tion of u Frequency expressed in wave-

number (cm-') Fast Fowler Transform (Cooley-

Tukey algorithm) Maximum intensity In interfero-

gram near 6 = 0 Deuterated Triglycine Sulfate-

type of pyroelectric detector Mercury-Cadmium-Telluride-a

liquid-nitrogen-cooled photo- conductive detector

Maximum value of optical retarda lion corresponding to maximum displacement of movable mirror

Full Width at Hall Height- a measure of re~olution using the method 01 observed band width

Instrument Line Shape

If the movable mirror is displaeed by a distance equal to one quarter of the wavelength of the emitting source (I = Al4) the

retardation becomes 6 = ,412. The two wave- fronts now reach the detector 180' out of phase, and we have destructive interference or cancellation. If the mirror is moved an additional quarter of a wavelength, or a total of AI2, the retardation then becomes 6 = A and once again we have reinforcement and s maximum sienal.

I t ran he seen fmm Rgure 1 that as retar. dation is increased the signal from the de. tector-the interferogram-goes through a series of maxima end minima. The maxima occur whenever the retardation is an integral multiple of wavelengths of the emitting source (6 = nX: n = 0. f 1. +2. etc.. .) .The minima occur 'when the ;et&dation' is an odd multiole or half waveleneths (6 = In + .. . , ! ~ ] h ) . The resulting interfenrgram can be dercrihed as an infinitely long cosine wave defined by the equation

I(a) = ~ ( u ) eos(2s:) (1)

in whieh I(6) is the intensity of the detector signal as a function of optical retardation and B(u) is the intensity or brightness of the source as a function of frequency. In infia- red spectroscopy it is more convenient to express wavelength in terms of the carre- soondine freauencv. and the unit of choice " . iswavenumber,decoedas v = l/h. With this substitution, eq 1 becomes

When the source emits more than one frequency, i t ispossible,at least in principle, to treat each freauencv as if i t resulted in a . . separate cosine train (each with its own pe- riodicity) and then to add the cosine waves geometrically to obtain the form of the resultant interferogram. This procedure is illustrated in Figure 2 for the case of a source emitting two closely spaced lines of equal intensity. Mathematically one may define the interferogram as the sum of the cosine waves of all the frequencies present in the source

"n

I(6) = B(P;) eos ( 2 ~ 6 ~ ~ ) (3)

The typical infrared source is a glower emitting a continuum which is usually treat- ed as a black bodv radiator, and the summa- tion is now replaeed by an integral

The interferogram from a typical radiating source is shown in Figure 3. Note that at zero retardation (6 = 0) all of the cosine waves from all of the frequencies present are in phase. Thus the signal I(6) always has a strong maximum st 6 = 0 and this feature of the interferogram is known as the centerburst. As we move outward in either direc- tion from the centerburst, the multitudi- nous cosine waves hegin to reinforce and cancel each other, and the intensity of the interferogram dies off rapidly into a series of lower amplitude oscillations. The less spectral structure in the source radiation, the more rapidly these oscillations die out. To put it another wily, the higher resolution information in the spectrum is contained farther out in the wings of the interferogram, corresponding to greater retardations and hence greater mirror travel. There is a ~ractical limit to how far the movable mirror can be displaced, and the optical retardation that corresoonds to this maximum dirplacement, repremwd by the symhol 3 r> = S,.,,, determines the theuretiral limiting resolution of the interferometer. The ex- pression for best resolution is

1 AD = -em-' A

( 5 )

In nractice this resolution is never attained berause of trunration or apodimtiun. There terms will be explained lawr.

Computation ol the Spectrum The interferogram is described mathe-

matically by eq 4, and it is the interferogram (intensity versus time) that is measured physically. The spectroscopist, however, is generally more interested in the spectrum (intensity versus wavenumber) of the radiation which produced that interferogram. If we know the mathematical form of the interferogram, I(6), as a function of 6, it is possible to calculate the corresponding spectrum using a mathematical technique known as Fourier Transformation. The ex- pression for the spectrum is

B(u) = I(*) cos (2su6)d6 (6) -

SPECTRUM INTERFEROGRAM

Fioure 2. The interferooram of a sinale freauencv source is a cosine function with a oeriodicitv that varies - . . wnhtne hequency ol me emin ng same (a and b) The nterlerogram 01 a tw~trequency source may be ca culated by geamencal y add nglnecor ne lunnmns correspomlngtocachoftne individual llnes nthe source (c)

A6 Journal of Chemical Educatlon

where B(u) is the intensity of the spectrum as a function of wavenumber ( u ) . To be mathematically valid the integration must be performed over all posQble values of retardation from minus infinity to plus infinity. The two equations 4 and 6 are known as a Fourier pair. Since the spectrum and the corresponding interferogram are related to each other by the Fourier Transformation, this technique of obtaining aspectrum from an interferogram is known as Fourier Trans- form Infrared Spectroscopy, ahhreviated FT-TR.. . .

While some emission soeetrwcoovis done ~ ~~ . . in thr infrared, most of the measurements made there are of absurption spectra. Note the sample position shown in Figure 1. If n,, sample is present, the spectrum is that of the emitting source and resembles the spectrum of a black body radiator modified by any transmission characteristics of the interferometer components (such as moisture protective coatings on the heam splitter or the detector window). If the interferometer chamber is not evacuated (this is rarely done) or purged with dry gas (usually done), some absorption from atmospheric water vapor and carbon dioxide will also he observed. This spectrum is known as the background spectrum or, simply, the baek- ground.

When a sample is introduced, the spectrum now exhibits all of the characteristic absorbance bands of the sample superim- posed on the rather uneven background (see Fig. 4). I t is in fact a single beam spectrum just like those produced by early dispersive spectrometers hefore the perfection of double beam recording. To obtain the more familiar mesentation of %T versus wavenum- brr we need to ratio the single beam sample sppctrum ngnrnrt the barkgruundrpectrum. This process ~s illustrated in Figure I where we see graphically the relationship between the background and sample interferograms, their transformed single beam spectra, and the resultant "double beam" ratioed spectrum.

Ratioine is done in a dedicated comnuter which is an integral pan of the FT-IH spectrometer. This rumputer is essential tocnrrs out the Fourrer Tran~furmation but is then also available for the ratioing process and far other spectroscopic data processing such as smoothing, reformatting, flattening, the taking of difference spectra, etc.

A sin& translation of the movable mirror rr adequate to produce an in te r f~ro~rnm that can bt. transtormrd into a spectrum. better signal-to-noiae ratios can he ahtained however by averaging a number of scans. Noise is reduced in proportion to the square root of the number of scans averaged, and i t is common practice routinely to average several scans so as to improve spectral quality. Since individual interferometer scans are usunlly made m a secund or less, this is much more fea*il,le fur FT-IH than for dirperriw spectroscopy where scans require a matter of minutes.

Since the final transmission spectrum is made by ratioing a sample spectrum against a background scan made a t a different point in time, i t is important that nothing about the system change between the scanning of the background and the scanning of the sample. This introduces requirements on source stability, mechanical stability of the optical bench, reproducibility of mirror mo-

Figure 3. lnterferogram from a typical inhared glower. Note the strong centekwrst at 6 = 0.

INTERFEROGRAM SPECTRUM

BACKGROUND

- FFT-

BACKGROUND (SINGLE BEAM)

INOENE (SINGLE BEAM 1 *

"DOUBLE BEAM* OR RATIOED SPECTRUM %T

Figure 4. The generation of a spechum by an FT-IR spectrometer. An interferagram of the source (background) is scanned, hansformed into a single beam spectrum, and stored in computer m e w (a). The sample, indene in an 0.025mm cell, Is placed in the beam and the process is repeated (b). The two singl-beam spectra, in computer m e w , are ratioed to producedthe more conventional "double beam" presentation (c).

tion, and repeatability of the level of purg- by scanninga background,storingit incom- ing. For the most precise results it is desir- puter memory, and then ratioing a number able to run a haekeround immeditelv Dre- of subseauent s a m ~ l e scans against the ceding the samplescan or, in the Ask of averaged scans, to interleave the background and sample scans. In practice however excellent quality spectra are obtained

.

(Co,rtinued on page AS)


stored background. A new background must of course he run if there is a change in resolution or apodization, or if a shift in hack- ground is noticed in the ratioed spectrum.

Design Varlatlons The most common configuration for an

FT-IR spectrometer consists of a glower or heated element as the source, a germanium beam splitter evaporated onto a KBr sub- strate, and a deuterated triglycine sulfate (DTGS) pyroelectriedetector. (The familiar radiation thermocouple commonly used in most dispersive instruments cannot he used in FT-IR because its response time is too slow for the rapid (one second or less) scan time used in the interferometer.) These components introduce certain limitations in performance. Potassium bromide does not transmit below about400 cm-1. We can scan to about 180-200 cm-I by changing to a CsI beam splitter. Below that limit unsupported Mylar films can be used and measurements can then be made almost to the millimeter region.

As a detector material DTGS has useful sensitivity over practically the entire infrared spectrum. An individual detector will he limited by the transmission of its window. The most common window, CsI, because i t is very thin, can be used down to about 150 cm-'. Below this frequency a polyethylene window must be used. When the signal is very low as is often the ease when working with an infrared microscope or examining gas chromatography fractions, alternate detectors are often employed. Of these the most common is the liquid-nitrogen-cooled mercury cadmium telluride (MCT) detector. This detector comes in a variety of forms in whieh transmission range is traded against sensitivity (D* value), and when an MCT detector is used, signal-to-noise ratios can be increased by as much as an order of magnitude.

The radiating source is perhaps the most limiting of the FT-IR system components. Thermal sources are universally used, and for most mectroscoov the" can he ooerated .. , at relati\cly low rrrnperatures. usun.ly 1000-i?OUD<.'. Radiant pvwcr dues fall 011 at ldnyrr wa\,elcngths according tu Plnnrk's law, and source energy becomes limiting as we approach the far infrared. At very low frequencies a mercury arc source is some- times used.

In the original Michelson design retardation between the two legs of the optical path was varied by translating the movable mirror parallel to itself. Any tilt or wobble of the mirror during the motion will distort the interferogram and subsequently the transformed spectrum. In addition, the velocity of the mirror must also he maintained pre- cisely constant since the detectors used in FT-IR have a frequency dependent response. These considerations place strin- gent requirements on the mirror drive system. One of the most widely used mirror drives utilizes an air bearing where themov- ing parts literally ride on a thin cushion of dry gas. A number of attempts have been made to simplify the drive system. In some designs the air hearing has been replaced by a mechanism best described as an old-fash-

ioned porch swing. Another variation, first used in the visible region of the spectrum, relies on the rotation of a pair of parallel mirrors rigidly mounted on a pivoting base plate. The transept design abandons the moving mirror entirely and changes path- length by driving a wedge of refractive material (such as KBr or CsI) into one leg of the optical path. These techniques simply rep- resent alternate ways of varying the retardation and are technically equivalent.

Apodlzatlon If one examines the mathem#tieal form of

the Fourier pair (eqs 4 and 6) i t can be seen that in order to obtain a complete spectrum, B(u), one must perform the integration over retardation values from minus infinity to plus infinity. This is of course impossible since mirror travel cannot extend to infinity but is limited to some finite distance, usually determined ultimately by the design of the mirror drive.

The consequences of this situation, and the oractical resoonse to them. are best un- derstood hy rrlurning to the ~implfiention 01' a single infinitely narrow emrtting linr xuurce. \\'e saw earlier that its interfenrgram consisted of an infinitely long cosine wave.

In the simplest concession to reality, if retardation ranges from -A cm to +A cm (see Fig. 5), this amounts to truncating the interferogram, or setting it equal to zero for all values of 6 greater than A and less than -A. In mathematical terms this is known as convoluting the complete interferogram (-- < 6 < + m ) with a fundion whieh has the value 1 for values of 6 between -A and +A and the value O for values of 6 outside

these limits. Because of its rectangular shape, this function is likened to a boxcar, and the process is known as hoxcar truncation. There is of course a price to he paid. We have lost high resolution information from the extremes of the interferogram, and when we do the Fourier Transformation with boxcar truncation, the infinitely narrow line takes on width. Mathematically the Fourier Transform of the boxcar function is of the form (sin x)/x which is known as a sinc function (see Fig. 6). The transform of the

Figure 6. Effect of boxcar truncation. The Fourier Transform of the boxcar function is a sinc function: sinc x = (sin x)/x In the transformed spectrum, an infinitely narrow line takes on width and acquires positive and negative sids lobes.

infinitelv narrow line now has a full width a t ~ ~~

half height rFWHH1 of O.fiO5.4 and is ar- wrnpanicd 11s ierres of aid? lobes, undulat- ing pusitivc and negatrve with diminishing intensities as we move away from the central frequency. This phenomenon is known as ringing and can he observed in the top spee- trum of Figure 7. The first minimum has a

Figure 5. Boxcar truncation. The imerferogram is set equal to zero fw values of 6 > A arm 6 < -A. Between -A and +A the interferogram is multiplied by a factor of 1.0 (i.e., lefl unchanged).

Figure 7. Spectra of HCi in a 10-cm gas cell run ate nominal resolution of 1 em-' with boxcar truncation (a) with triangular epodizatian (b). Resolution is higher for boxcar truncation but the spectrum exhibits "ringing" (i.e.. many sids lobes are presem). The side lobes are eliminated by triangular apodization but re~olution is lost.

A8 Journal of Chemical Education

negative d u e uhieh is ahout 2% of the nmplitude d the central positive frequrnrv. If a second weak line were present in the source at the frequency of this minimum, it could be last from the computed spectrum.

In order to avoid the problems from negative side lobes we can substitute a triangular shaoed function far the boxcar function (see Fig. 8). The Fourier Transform of the triangular function is a sincZ function and has the form (sin%)/$ (see Fig. 9). The amplitude

Figure 9. Effect of triangular apcdlratlon. me Fou- rier Transform of the triangular function is a dnc2 function: rlnc2 x = (sin2dl.?. Compared to boxcar truncation. the transform of an infinitelv narrow me5 takes an addmna wdth ldecreawd reoolu- t~onl, o~ the magnrtude of the s de iaoes is redmed and they are now ail positive,

of the side lobes is considerably reduced and they now have all positive values. Once again a price is paid. Now the transform of an infinitely narrow line has a FWHH = 0.88/A, a significant broadening over the case of boxcar truncation. In some ways the triangular function is preferable to boxcar truncation since it results in the same instrument line shape (ILS) as a diffraction- limited grating spectrometer.

Note the spectra in Figure 7. The top spectrum was calculated with boxcar truncation and exhibits considerable ringing while the bottom spectrum was calculated with a triangular function. The triangular function has decreased the amplitude of the side lobes, or to put i t another way, "cut off the feet". Functions which reduce the side lobes are referred to as apodization functions, derived from the Greek, aro6os, meaning without feet.

The foregoing discussion is summarized in Figure 10 where we see the interferogram of an infinitely narrow line-an infinitely long cosine train. The spectrum calculated from this infinitelv lone train returns the . .. infinitely narnlw h e . \I'hen wc limit mirror trawl iretnrdauon, and w e boxcnr rrmcn- tim-rr is nut, strictly ,peaking, corwct to use the term hoxcar apodization since side lobes are not reduced -the calculated spectrum consists of a single broadened line with nositive and neeative side lobes. Resolution is the best obtainable, but baseline irregularities are also greatest. If we use triangular apodization, we further broaden the calculated line but significantly reduce the side lobes or baseline irregularities. One is tempted to make a comparison with varying the slit width on a dispersive spectrometer where narrow slits are associated with high resolution and higher noise levels and wider slits to lead to poorer resolution hut increased siensl-to-noise ratio. ~~~~~~~

~ - I t wvuld be incorrect to assume that our

only choices are hoxrar trunratiun and rri- angular npudinmm. Many uther functions have also been proposed and are used. A common alternative is trapezoidal apodiza-

Figure 8. Triangular apcdizatlon. The interleferogram Is set equal to zero for values of 6 > A and 6 = -A. Between -A and +A me interferogram is multiplied by a linear function which has me valve of 0 at -A and +A end a value of 1.0 at 6 = 0.

INTERFEROGRAM + FFT-

INFINITELY LONG I

COSINE WAVE s=o

8'-A S=O BOXCAR TRUNCATION

S=-A S ~ O 8 : ~ TRIANGULAR APOOIZATION

SPECTRUM

lY

Flgm 10 The llgure summarues lhe eHec1 of the mathematical treatmem of me lntsrferogram on the translarmed spectrum The source is an nflmteiy narrow ions Tne onlerferogram la) 18 an inlln tely long C O S ~ va8n *h chtr~nsf~rms falthfu y toreturn the nfin~tely w o w I ne Wnen boxcar truncatoon s "sea (b) the transformed line takes on width and exhibits positive and negative ride lobes. Triangular apadira- tian (c) further increases line width but reduces the magnitude of the sick lobes and makes lhem ell positive.

tion. The shape of the trapezoid may be varied by changing the length of the side parallel to the base. (The base extends from -A to A,) I t can he seen inFigure 11 that the trapezoid degenerates into a boxcar when the opposite side becomes equal to the base and into a triangle when the opposite side becomes a point. It is not surprising then that the resolution and side lobes in the spectra calculated using trapezoidal apodization are intermediate between those calculated using the hoxcar and triangular functions.

Still other apodization functions employ a degree of nonlinearity or curvature in an effort to maintain resolution while reducing the rinaina These include Gaussian and ~ o r e n t z i a n functions (seldom used), a raised cosine function (also known as Han- ning apodization), Happ-Genzel apodization, and Norten-Beer apodization. Of these, the Narten-Beer functions (7) (they are actually a family of functions) seem cur-

rently to he the most widely used. They are believed to offer the best compromise in maintaining resolution and line shape while at the same time reducing baseline irregu- larity. Most commercial FT-IR spectrometers offer some choice of apodization function and one software package even of- fers the user the capability of introducing an arbitrary function of his own choice.

This concludes part I of this paper, in which we have addressed the design of the FT-IR spectrometer, the computation of the spectrum from the interferogram, and the use of apodization. In part I1 we will discuss the advantages of FT-IR over dispersive techniques and show applications of FT-IR to difficult spectroscopic measurements.

(Continued on page AlO)


Figure 11. Trapezoidal apodization functions (b and

c). Note that the trapezoid degenerates into a boxcar function (a) when Me opposite side equals the

base and into a triangular function (d) when the opposite side shrinks to a point.

Literature Cited (1) Michelaon, A. A. Phil.Mag. 1691,31 Ser. 5,258. (2) Miehelson, A. A. Phil.Mog. 1892,34 Ser. 5,280. (3) Lord RayleighPhil. Mag. 1692.34. Sex. 5.407. (4l Feiigett. P. I n "Aspen Int. Conf. on Fourier S p t . ,

1970"; Vanasae, O: Stair, A. T.; Baker. D., Ms.; AFCRL-71-WI9 Cambridge MA. 1971: p 139.

(5 ) Cwiey, J. W.: Tukey, J. W. Moth. Cornput. 1966,19, 297.

(6) Forman, M. L. J. Opt. Soc. Amer. 1966.56,978. (7) Norton, R. H.; Bssr. R. J. Opt. Soc. Arnm 1976, 66,

259.

General References Horliek, G. "Introduction to Fourier Transform Speetroa-

COPY", Appi. Specfroseopy l968,22,617. Griffiths, P. R. "Chemical Infrared Fourier Transform

Spectroscopy"; Wiley: NewYork. 1975. Vanasse. G.; Stair, A. T.; Baker, D..Ms. "Aspen lnterna~

tional Conference on Fourier Spectraacopy"; AFCRL- 71-W19: Cambridge, MA, 1971.

Bracewell, R. "The Fourier Transformation and itr Appli- cdions": McGraw-Hill: New York. 1965.

A10 Journal of Chemical Education

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