Instrumentation
Alan G. Marshall and Melvin B. Comisarow Department of Chemistry University of British Columbia Vancouver, Canada V6T 1W5
Fourier and Hadamard Transform Met hods in Spectroscopy To understand why Hadamard and Fourier methods have proved so valuable in spectroscopy, it is first necessary to recognize the disadvantages of the conventional way of doing spectroscopy, in which the spectrum is obtained by scanning across the spectroscopic 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 unknown objects, by use of the schematic balance shown in Figure 1. Conventionally, 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:
Measurement =1 =2
=3 =4
No. ofweighings
=1
Unknown z2 3:
1
0
0
0 0 0 1
1 0 0 1
0 1 0 1
=4 0
0 0 1 1
The obvious advantage of this procedure is that the measurements yield each unknown weight directly; therefore, 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 imprecision or random noise, it is desirable 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 average absolute distance away from zero
Flgure 1. Schematic diagram of double-pan balance, set of standard weights, and four unknown weights after N steps of a random walk (more precisely, the root-mean-square distance) is proportional to v’T ( I ) . Thus, the true measure of precision of the repeated measurement, the signalto-noise ratio, is proportional to ( N l or just 0. Let us return to the balance problem. Suppose that two unknown weights are placed on the left pan at ’once and that four linearly independent arrangements of unknowns are weighed, two a t a time:
a)
Measurement 21 =2 =3 24
No.ofweighings
Unknown =1 =2 =3 1 1 0 0 1 1 0 0 1 1 0 0 2 2 2
=4
0 0 1 1 2
This time, the fdur unknown weights are related to the four observed weights by four linear algebraic equations, which may then be solved to yield the desired unknown weights. However, since each unknown has now been weighed twice, the precision (signal-to-noise ratio) of each calculated 2 than for weight is now better by d the original conventional experiment.
Since the same total number of weighings (four) are required, the total time required for the new experiment is also the same. For an arbitrary number, N,of unknown weights, the general improvement in signal-to-noise ratio for this encoding-decoding scheme is m 2 for an experiment that takes no longer to carry out than N conventional single-object weighings. [The precision of the calculated weight will be better only if the average error in a particular weighing depends upon the balance and is independent of the magnitude of the measured weight. This general condition is called “detector-limited” noise and is distinguished from the “source-limited” 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 encoding-decoding method will not improve the precision of the calculated weights over those determined in a simpler one-at-a-time weighing procedure (2). Spectroscopic examples in which the noise is detector limited include infrared, microwave, nuclear magnetic resonance, and ion cyclotron resonance experiments. Examples in which the noise is source limited include optical (VIS-UV) and charged-
ANALYTICAL CHEMISTRY, VOL. 47, NO. 4 , APRIL 1975
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particle (photoelectron, ESCA, electron impact) spectroscopy.] Although the experimental details of the spectroscopy experiment are completely different from those of the balance experiment, the preceding encoding-decoding scheme still applies and forms the basis for Hadamard transform spectroscopy, as will shortly be evident. By logical extension of the preceding argument, one might think of putting 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 unknowns: Measurement
Unknowns in Right pan Left pan Z1 :2, 23, =4 1: =1,,12 *3,:4 22 31, *3 *2, 4: 23 =1,44 $2, ,13 24 (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 improved by 4 = 2 times over conventional 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 This improvement also applies to the (experimentally different) Fourier transform spectroscopy experiment, 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 v'7V72 or 4 V , respectively, but require the sume total time for measurement. This improvement is known as the Fellgett advantage (3);all that remains is to show that we ordinarily perform spectroscopy as inefficiently as we ordinarily use a double-pan balance, and we then explore the available means for exploiting the potential Fellgett advantage in the situation. [The Fellgett advantage can be realized only if the noise is detector limited, not when the noise is source limited (21.1
a.
Direct Multichannel Spectrometers Figure 2' 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 492A
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
Figure 3. Schematic diagram of detector section of direct multichannel spectrometer composed of many separate single-channel detectors
that when detector readings are collected 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 spectrometer is that its detection of an absorption spectrum requires a procedure formally identical to the one-at-a-time method of determining the weights (spectral intensities) of N different unknown objects (spectral slit positions). I t would thus be desirable to open up the slit aperture to the full width of the desired spectral window by using N separate single-channel detectors as shown in Figure 3. Since all slit positions are now monitored a t once, rather than just one a t a time, the spectrometer of Figure 3 offers (in principle) an improvement of the full 0advantage 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
ANALYTICAL CHEMISTRY, VOL 47, NO. 4, APRIL 1975
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 a t a time. Because of the conceptual simplicity of the spectrometer of Figure 3, it is logical to investigate its feasibility. I t 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 Table I. From Table I, it would appear that electronic (VIS-UV) spectroscopy is the least likely candidate for success with a direct multichannel spectrometer; but in fact, multichannel detection of optical-UV radiation is readily accomplished photographically. The
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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.
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 photoelectron 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 electronic 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 detector (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 detector channels (see Table I), the Figure 3 spectrometer is thus now feasible for ESCA and photoelectron spectroscoPY. With the other forms of spectroscopy listed in Table I, direct multichannel methods are less attractive. For microwave (rotational) spectroscopy, for example, there is no broad-band radiation source available: a blackbody radiation source, such as employed for other radiation energies (xenon or hydrogen discharge for UV, hot tungsten wire for visible, globar for near infrared and infrared, mercury vapor for far infrared) would have to be operated at an unreasonably high temperature to obtain sufficient 494A
radiation flux for use as a radiation source. It would be conceivable to construct an array of individual (narrowband) microwave transmitters (about $5,000 each) as the “broad-band” radiation source, but Table I shows that the cost would be excessive ($IO9 . . . !). For infrared spectroscopy, on the other hand, the necessary broadband 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 existing individual (thermopile) detectors of about 1-mm width each. At a cost of about $200 per detector, the total cost again becomes unmanageable. (Photographic detection does not extend beyond about 12,000 8, 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
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: XI, x 2 , . . . , 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, a,, of either zero or one, depending on whether that particular slit was shut or open, respectively: y = alxi
+
a2x2 +
,
..
+
In other words, the detector has provided one observable ( y ) ,expressed in terms of N unknowns (XI to X N ) ,according to the “code” (a1 to a ~of ) Equation 1. This situation is clearly parallel to that of putting half the un-
Figure 4. Disperser, mask, and (single, broad-band) detector of Hadamard spec-
trometer
ANALYTICAL CHEMISTRY, VOL 47, NO. 4, APRIL 1975
Incoherent and Coherent Spectrometers
Figure 5. Functional equivalence of four separate masks (N possible slits each) and movable single mask [(2 N - 1) possible slits]
just one, while hoth experiments require the same length of time for execution of N separate intensity measurements. [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 eauations. such as Equations 2 , i y a digital computer requires that the matrix of the coefficients he “well conditioned” (7). Fortunately, the matrix of coefficients generated by the scheme of Figure 5 can readily be transformed to such a well-conditioned form, and Equations 2 can thus he solved accurately.] Alternatively, the Hadamard spectrometer can provide the same signal-to-noise ratio in a aiNxN factor of (2/N) as much time as would be required by the conventional sina2x ,, gle-slit spectrometer. Hadamard encoding-decoding aNNxN methods have been applied most prominently in infrared spectroscopy (2) (8, 9).
known weights on the left pan of the balance, as already discussed. To recover the desired spectrum, x i 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 approximately half the slits open, so that this second slit arrangement is linearly indeDendent from the first. Proceedine in this way, one readily obtains N observables (the total transmitted intensity through each of N different masks, y1 to Y N ) , expressed in terms of N unknowns (the spectrum, XI to IN),according to a “code” in which all the coefficients are kither zero or one, and roughly half the coefficients in any one row are zero:
yi = ail,%, + ai2x2 y2
I
yN
+ a,x,
+ . . .+
+ . . .+
aNixi + aN2x, +
. . .+
A particularly convenient method for providing the N linearly independent slit arrangements is shown in Figure 5. I n s t i d of N separate masks, it suffices to construct just a single mask consisting of (2 N - 1) potentially open slit positions: then by opening the window over just the first N positions, one constructs the first slit arrangement a t the left of Figure 5 ; by opening the window over the second through ( N 1)positions, one creates the second slit arrangement a t the left of the figure, etc. Mechanically, the change from one slit arrangement to another simply consists of translating the linear (2 N - &slit mask across the window by one position per change. On hoth computational and mechanical grounds, the Hadamard approach of Figures 4 and 5 conveniently achieves an improvement of a factor of vW2 in signal-to-noise ratio over the conventional one-slit-at-a-time scanning spectrometer of Figure 2, because half the N possible slits are open during each measurement, rather than
To proceed to Fourier methods, i t is important to understand that the spectrometers discussed up to now (Figures 2-5) can operate with an incoherent 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 coherent source; the coherent source makes possible another type of encoding-decoding scheme (Fourier) for opening the spectral window while preserving resolution (Figure 6). Finally, one reason that hoth Hadamard (incoherent source) and Fourier (coherent source) methods can he applied to infrared spectroscopy is that the Michelson interferometer can he thought of as a device which effectively converts incoherent to coherent radiation in the present context. A coherent radiation source and coherent detector in a spectrometer provide two important advantages to the spectroscopist. First, since the frequency of the coherent radiation source is easily determined to very high accuracy by use of an electronic counter, the line positions in a spectrum may he determined very accurately, simply by measuring the frequency of the source as it is (slowly) scanned over the spectral window.
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Figure 6. Spectroscopy classifiedby radiation source and spectrometer bandwidth
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Figure 7. Hypothetical infrared laser source spectrometer
Second, coherent radiation permits the implementation of electronic filtering techniques which can make the spectrometer resolution arbitrarily high. Thus, the spectral lineshape determined by a coherent radiation spectrometer can be made characteristic of the sample, by making the instrumental broadening arbitrarily small. The basic operation of a coherent source spectrometer is shown in Figure 7, consisting of a hypothetical infrared laser source spectrometer for use in vibrational spectroscopy. The radiation issuing from the source consists of a plane-polarized electric field whose magnitude varies sinusoidally with time. Upon encountering an electric dipole (Le., a polar molecule), the electric field will force the dipole to oscillate a t the frequency of the radiation, 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 coherent, then all the dipoles in a given region of space will oscillate together, forming a macroscopic oscillating electric dipole in the sample. That macroscopic oscillating dipole then induces an oscillating charge (and thus a corresponding oscillating voltage) on the parallel plates of the capacitor enclosing the sample in Figure 7 . That induced 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 decomposed algebraically (by a trigonometric 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 498A
spectral segment which is centered a t the source frequency and whose width is determined by the bandwidth of the low-pass filter. For example, the transient 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 position which is determined by the frequency 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 arbitrarily wide or narrow without any mechanical adjustment of the spectrometer geometry. Spectrometers in which a macroscopic change in a physical property of the sample is induced by radiation from a coherent source, and that macroscopic change is detected electronically, in the manner described above, have long been employed in NMR spectroscopy ( 1 0 ) and ion cyclotron resonance (Figure 8) spectroscopy ( 1 2 ) and have recently been introduced in microwave (12) and infrared (13)spectroscopy.
Fourier Transform Spectroscopy Fourier transform methods a t 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. It is, however, equally useful to analyze observations in the time domain (Figure 9). The upper left diagram of Figure 9 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 at zero frequency, but the actual frequency representation 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 91,the frequency representation is a completely flat spectrum. [The mathematical correspondence between the time domain and frequency domain diagrams of Figures 9 and 10 is called a (0-ne-sided) Fourier transformation ( 2 4 ).] T h e diagrams in Figure 9 suggest that the broad-band frequency excitation required for a multichannel or multiplex spectrometer can be generated by use of a sufficiently narrow electromagnetic radiation pulse. (If the pulse consists of an AC rather than a DC waveform, then the pictures of Figure 9 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 Figure 10.) For NMR, for example, Table I indicates that an excitation bandwidth of about 10 kHz is required-Figure 10 (top trace) indicates that such an
Figure 8. Schematic diagram of ion cyclotron resonance (ICR) spectrometer
ANALYTICAL CHEMISTRY VOL 47. NO 4 . APRIL 1975
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.hand, the oscillation is observed for several lifetimes of its decay (middle trace of Figure lo), the spectral representation approaches the familiar Lorentzian line shape encountered in many forms of spectroscopy. Finally, the bottom trace of Figure 10 illustrates the intermediate case in which the acquisition time, T,is of the order of the decay lifetime, 1. The irreversible 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 interaction may he neutral-neutral collisions (microwave, infrared, optical); ion-molcule collisions (ion cyclotron resonance); rotational diffusion (nuclear magnetic resonance, electron spin resonance); or depletion of the excited sDecies owine to chemical reaction. The multichannel advantaee of the Fourier approach can now he understood. Suppose that the time domain response, y ( t ) , is sampled a t N equally spaced intervals during a total acquisition time, T.Each of these sampled time domain points, y(t,,), n = 1to N, is then a linear combination of all the discrete frequency domain spectral points, x(w,). according to: I
I
Figure 9. Time domain (left) and frequency domain (right)representations of DC pulses of three different durations
Y(tJ =
aiix(Wi)
+ a,,x(w,) +
. ..
+
UiNX(WN)
Y(t2) = a*tx(w,) + a22x(w2)
+
. .. +
d t N )
=
aZNx(wN) aNiX(wI) + aN2x(w2)+
.. .
+
aNNx(wN)
(3)
i n which a,,, = exp [Z n i m t , / ~ ] o r just
(4)
a,, = exp [2 nimn,”]
(5)
Equations 3 should he compared to Equations 2 since there are now N in. . . . dependent observed sampled time doFigure I O . Frequency representations (right)of three types of time domain (left) main points, y(t1) t o y ( t ~ )each , exspectrometer signals pressed in terms of all N discrete frequency domain points, ~ ( w to d ON), it is again possible to “decode” the obexcitation may be produced simply by When a given single oscillator is served data to obtain the desired specapplying a radiofrequency pulse whose trum. The decoding procedure, called subjected to irradiation a t its resonant duration is of the order of 10 Fsec. As frequency, the amplitude of oscillation a discrete Fourier transformation, will increase. If the irradiating excitaanother example, electron impact may he calculated rapidly by a digital spectroscopy (15)is based on the computer (18).In contrast to the tion is then removed, the oscillation will persist with an amplitude which rapid passage of an electron past a Hadamard technique, in which half decreases (usually exponentially) with molecule. This passing electron prothe possible spectrum is detected in duces a very short, sharp pulse of electime, as shown for three convenient any given observation (Le., half the trip h.w limit.ino rit.nat.iona at. n-2 . ...fieid ... .-at. -.t.ha ....rnolamle .... .. . -..and- t.. -.arts -. . . . . . . . . . . . ~ I -.the .... left. . . . . of -.Fivnre ..~-.-,.,,.in ... anv -..,one . ...Tow .... of . .lhnations - m---e as a very broad-hand, nearly flat 10. If the oscillation is not appreciably zero), the magnitude of each ann in source of irradiation. In this case, the reduced during the acquisition time, T the Fourier experiment of Equations 3 is unity: frequency bandwidth is sufficient to (top trace of Figure lo), then the corexcite the same sort of transitions as responding frequency representation are more conventionally studied in has a functional form which resembles lan,,, = [ e x p [2 n i n m / ~ ]= / 1 photoelectron and ESCA spectroscothe amplitude of (Fraunhofer) diffraction by a slit (16).If, on the other (6) PY. 500A
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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, i t is now clear that detection of the time domain response, followed by Fourier transformation to obtain the frequency domain response, provides a frequency spectrum exhibiting either signal-to-noise improvement of a factor of flin the same total observation period; or a spectrum having the same signal-to-noise ratio in a factor of (11 N) as much time as required by a conventional 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 transformation of the (spatially dispersed) sampled interferogram (19). Since half the spectral intensity is necessarily lost a t 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 interpreting frequency spectra obtained by Fourier transformation of a time domain response is acomparison to the spectra obtained by a conventional slow-sweep spectrometer. Under the very generally valid condition that the system response be linear Le., proportional to the magnitude of the irradiating 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 components 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 a t a “cyclotron” frequency, u = qB/2nm, [mks units; v in Hz]
If such ions are placed between two parallel plates, as in Figure 8, and irradiated with a coherent (circularly polarized) electric field whose frequency is close to the ion cyclotron frequency, the resultant ion motion can he shown to become spatially coherent as the ions ahsorh energy from the irradiation by increasing the radii of their cyclotron orbits. Once the ions are all moving essentially together, their composite cyclotron motion will induce a macroscopic voltage in the sur502A
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ANALYTICAL CHEMISTRY, VOL
Flgure 11. Transient ICR signal from sample of N2+ and CO+ rounding plates, and that voltage signal may be amplified and recorded as shown in Figure 8. The main conceptual difference between the ion cyclotron 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 1MHz, as indicated a t the bottom of Figure 8. It is now apparent that an ICR spectrometer provides a signal whenever 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 irradiating the ions with an oscillating electric field whose frequency is scanned over the ranee reauired bv: Y ~. = qBWam. Fieure 11shows the ion cvclotron resonance ( I C H i rime domain response following excirarion of a bnndwidth sufficient to excire ion cyclotron resonanre for both h . 2 - and COT. Sincc the masses (and thus the cyclot r m frequencies) of S?- and CO- differ slightly, the time domain response of Figure 11 is a superposition of two decaying sine waves, whose heat pattern is evidenr in rhe figure. Fourier transformation of the time domain data of Figure 11 yields the ICH frequency spccrrum shown in Figure 12. Based on these and other (20-231 prororype experiments, Fourier techniques promise to reduce by a factor of 1.000 the time required to obrain an ICR mass spectrum. ~
Conclusions The value of any instrumental improvement must be g3uged on the hasis of its impact in making possible new experiments ior experts and better routine measurements for nonexpertc. On this hasis, Fourier methods 4 7 , h O 4 , APR L 1975
I
2AM
28.00
= 70,000
MASS ( I . r n Y . 1 -L
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 possible 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 Fourier data reduction in NMR), for example, carbon-13 NMR spectra were obtainable only with great difficulty. Today (1975), most major chemistry departments use carbon-13 NMR spectra routinely in structural and kinetic analysis because i t 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 (26) and NMR (27) spectroscopy, the recent application of Fourier techniques to electrochemical (281, microwave (12),ion cyclotron resonance (20-23), dielectric (291, and solid-state NMR (30) phenomena promises to make available to practicing chemists a broad new range of ex-
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Sensitivity. 1 ppm to 100% water with one buret. Precision. 1% accuracy, 0.5% reproducibility. Color-coded positive indication of end point. Rapid. 2-minute determinations. Uses 4 oz. disposable jars for reaction vessels; avoids delays between runs. Detect 3 nanoliters of water with L/l’s 2 ml micro vessel, interchangeable with 4 oz. jar. Applications. Unlimited. Includes petroleum products, pharmaceuticals, freeze-dried products, waxes, dehydrated materials, cosmetics, detergents, plastics, foods, wood products, paint, explosives, and many more. Price. $495.00 complete. Meter electrode alone, $1 52 Many accessories available, including Grunbaum pipet for standardizing with pure water. Contact L / I for free brochure and price list.
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”With the Grumman ADP-30SD we now process three times as many samples without increasing our costs’.’
periments not previously accessible. Ordinarily, the details of experimental measurement, although 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 a double-pan balance should best be used, it is possible to encompass a wide spectrum of spectroscopic applications of direct chemical interest. References (1) D. F. Eggers, Jr., N. W. Gregory, G. D.
Marion Walker, Health MaintenanceCenters,Inc., o subsidiaryof the American Health Corp., NewYork ”Here at Health Maintenance we’ve had o tremendous increase in our laboratory load.
Inoneyearwewentfrom300talOOOtriglyceride 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 ADP30SD dilutor/dispenser. “It has now been working for us for eight manths.That comes to about 32,000 tests. Errors have been significantly reduced, and I haven’t had to increase our costs. An equivalent yearlysaving in timeof over $10,000. “No wonder I think theADP30SD is well worth the money we invested:’ For information and a free triol of the ADP-30SD, write Grumman Data Systems Corporation, Instruments Systems Products, 45 Crossways Park Drive, Woodbury, New York 11797. Orcall (516) 575-3888.
Grumman Data Systems
Halsey, Jr., and B. S. Rabinovitch, “Physical Chemistry,” p 392, Wiley, New York, NY, 1964; W. J . Moore, “Physical Chemistry,” 3rd ed., p 232, PrenticeHall, Englewood Cliffs, NJ, 1962; N. Davidson, “Statistical Mechanics,” p 283, McGraw-Hill, New York, NY, 1962. (2) L. Mertz, “Transformations in Optics,” p 9, Wiley, New York, NY, 1965; R. J . Bell, “Introductory Fourier Transform Spectroscopy,” p 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. 0. Werme, R. Manne, and Y. Baer, “ESCA: Applied to Free Molecules,” 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 example. (7) G. Forsythe and C. B. Moler, “Computer Solution of Linear Algebraic Systems.” Prentice-Hall. Endewood Cliffs, N J , 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 Resonake,” 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., Vol2, Heyden and Son, New York, NY, 1971; J. M. S. Henis in “Ion-Molecule Reactions,” J. L. Franklin, Ed., Vol2, Plenum, New York, NY, 1972; G. A. Gray, Adu. Chem. Phys., 19, 141 (1971); C. J. Drewery, G. C. Goode, and K. R. Jennin s, in “ M T P International Review of icience, Mass Spectroscopy, Physical Chemistry,” Series One, A. D. Buckingham and A. Maccoll, Eds., Vol 5 , p 183, Butterworths, London, England, 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, p 152, Academic Press, New York, NY, 1973. (12) J. C. McGurk, T . G. Schmalz, and W. H. Flygare, J . Chem. Phys., 60,4181 (1974). (13) R. G. Brewer and R. L. Shoemaker, Phys. Reu., A6,2001(197‘2);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 P . C . von Thuna. Nature, 253, 514 (1975). (14) R. Bracewell, “The Fourier Transform and Its Applications,” p 360, McGraw-Hill, New York, NY, 1965. (15) C. E. Brion. in “ M T P International Review of Science, Mass Spectroscopy, Physical Chemistry,” Series One, A. D. Buckingham and A. Maccoll, Eds., Vol5, Chap. 3, Butterworths, London, England, 1972. (16) W. H. Furry, E. M. Purcell, and J. C. Street, “Physics for Science and Engineering 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 E. B. Wilson, Jr., “Introduction to Quantum Mechanics,” 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). 1211 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 andA. G. Marshall, J Chem. Phvs.. 62, 293 (1975). (24) P. Connes, Annu. Rev. Astron. Astrop h y s . , 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. Johnson 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, A p p l . Spectrosc., 27,85 (1973). (27) R. R. Ernst, Adu. Mag. Reson., 2, 1 (1968 1. (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. Griffin, 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.
