Instrumentation
Derivative Spectroscopy with Emphasis on Trace Gas Analysis Robert N. Hager, Jr. LSI/Spectrometrics, Inc One lnverness Dr. Englewood, Colo. 801 10
Derivative spectroscopy can extract more information contained within a radiation intensity spectra 1 distribution t h a n is accessible through direct spec tr osc opy techniq ues because of t h e specific measurement of the derivatives of that distribution As science probes deeper into any field, the advance must be accompanied by a capability to make new measurements, and often this advance of science forces the creation of new measurement techniques. A new measurement technique may be useful only in one narrow application and, therefore, of no general interest, but more often the technique may have broad applications quite beyond the original intention. Such is the case for a technique called derivative spectroscopy, which will be discussed throughout this article. The text will emphasize the application to trace gas analysis, not because the technique is restricted to this application, but.because detailed studies have carried this application from theory to hardware. Derivative spectroscopy was introduced as early as 1955 ( 1 ) as a means for resolving two spectral lines of very nearly equal wavelength. To accomplish this resolution, the first or second derivatives of a n intensity signal were obtained electronically (2) as a function of time, wherein wavelength was changed linearly with time. The result was a signal similar to the first
or second derivative of intensity with respect to wavelength, thus measuring the slope or curvature in the intensity distribution. A measurement of slope or curvature focused on the more subtle characteristics of the overlapping spectral lines, and thus the two unresolved lines could be separated. This unique feature of resolution was utilized in other specific applications and reported in the literature during the sixties, and derivative spectroscopy was born (3,4 ) . Derivative spectroscopy, then, is applicable to situations where information can be gained by looking for features in a spectral distribution. This automatically makes the information gained more specific. Any spectral distribution can be defined
more and more accurately at a given wavelength if, in addition to knowing the intensity at that point, the derivatives are also known. This principle is expressed mathematically by the Taylor series. As higher and higher derivatives are known, the more accurately the function is defined. If measurement information is contained in a spectral distribution, each higher derivative of the intensity contains more specific information. For example, if the intensity of radiation at a particular wavelength is measured after the radiation passes through a gas sample, a decrease in intensity indicates the presence of an absorbing compound. There may be many gases which could absorb this radiation. but information as to the
Figure 1. Intensity distributions a n d their derivatives ANALYTICAL CHEMISTRY, VOL. 45, NO. 13, NOVEMBER 1973
1131 A
specific one is lacking. If the first derivative is measured at this wavelength also, information is added. A constant first derivative (slope) indicates that the absorption varies linearly with wavelength in this region. This information can decrease the number of potential candidate gases. If the second derivative is measured and increases, the increase in local curvature indicates an absorption line or band a t this wavelength which greatly restricts the possible candidate gas. So, in this example, derivative spectroscopy supplies information which provides specificity to the measurement. This example is illustrated in Figure 1.The first and second derivatives are shown for two different spectral distributions. The dotted line, I,, illustrates the intensity before passing through an absorbing medium; the solid is the distribution after some radiation is absorbed. Note that the amount of light absorbed at A, is identical for both cases. In one case, the medium absorbs dominantly over a narrow wavelength region; in the other case, absorption is broad band. Note that if the intensity a t A, alone was monitored, no discrimination could be made between the two compounds. However, as the first and second derivatives are evaluated a t A, the presence of the characteristic narrow band absorption of the first case becomes evident. When only the second derivative is monitored, the two cases are easily separated since t h e broad band absorber produces no signal. It is true that intensity alone can be measured as a function ofwavelength by using a direct spectrometer. The information on slope and curvature is then within the output d a t a but not readily accessible. It is not a n “observable.” Intensity is the observable, the signal from the instrument, and this must be processed again to gain information on slope or curvature. This processing adds inaccuracies and sensitivity limitations which are set by external equipment and not by the original instrumentation setup. The signal output of a derivative spectrometer, the observable, is the value of the derivative. If this observable can be related directly to desired information, then accuracy and sensitivity can often be substantially improved.
gases. To look a bit deeper into this application, consider the relation between the derivatives of intensity and gas concentration. Starting with the function that contains the information, the intensity distribution is predicted by Beer’s law:
A
= In
IJ I
-
=
abc
where A is absorbance, a is the absorption coefficient as a function of wavelength for a particular gas, b is the pathlength of radiation through the gas, and c is the gas concentration. The derivatives of intensity can be influenced by a ( A ) and I,(A).although the latter can be selected somewhat to fit a desired situation. T h a t is, the radiation source can be chosen to produce only minor contributions in the derivative signal within a working wavelength region. Then, the first derivative
dI dI, d~ e x p (-abc) ; i ~= ; da bel, exp (-abc) or
a
The two terms in the above equation are independent of intensity. The first is a constant indicating the amount of slope in the radiation source distribution. The second term varies linearly with gas concentration. Two features of the above equation are noteworthy. First, with a derivative spectrometer, a signal can be obtained which is linear with concentration. This is more desirable than the logarithmic relationship by use of direct spectroscopy. Secondly, the sensitivity to concentration of the signal depends now on a physical property, the rate of change in absorption coefficient a t a particular wavelength, da/dA.This fact may or may not be useful in a given application, but it certainly introduces a new special class of compounds that would be applicable to the analysis technique. The same mathematics can be carried a step farther to obtain the second derivative of intensity with respect to wavelength. The result is a bit more complex and produces four terms as follows:
Theory of Derivative Absorption Spectroscopy
The preceding example of radiation absorption was not chosen a t random, for much work has been done in the application of derivative spectroscopy to the analysis of trace 1132 A
(1)
The first term is a constant measuring the curvature in the radiation source. The second and third terms
ANALYTICAL CHEMISTRY, VOL. 45, NO. 13, NOVEMBER 1973
may be useful for certain applications but ruin linearity for analysis work. However, note that a proper choice of wavelength can eliminate both terms. For analysis purposes, the wavelength chosen for measurement should be exactly where an absorption band occurs. At this point, curvature will be a maximum. Also, a t this point of maximum curvature, the slope is zero, as shown in Figure 1. Therefore, the second and third terms reduce to zero at this wavelength, and again a linear measurement with concentration is obtained from the last term. Kow, t h e sensitivity factor is the curvature in absorption coefficient which further discriminates those compounds which will create a secondderivative signal. Source of Derivative Signal
A true derivative spectrometer obtains t h e derivative signal optically, not electronically. This is extremely important, for a n intensity signal which is conditioned electronically, in fact, takes the derivatives with respect to time. The wavelength is varied linearly with time so that the result looks like a derivative of intensity with respect t o wavelength. However, time fluctuations in the intensity signal (noise) will be sensed, and the resulting derivative of the noise will be noisier. If a technique senses onl3, changes in intensity with wavelength, time fluctuations will not be sensed and noise is minimized. The optical technique used in a derivative spectrometer is u a c e i e n g t h modulation. The wavelength of the radiation passing through the sample is modulated sinusoidally with time, the wavelength amplitude of modulation generally being the width of the spectral feature to be analyzed. Consider Figure 2 . An intensity distribution is shown which contains slope (first derivative). If radiation is modulated about A, in time, a modulation in intensity will be induced. This induced modulation jrequenc?, is equal to the wavelength modulation frequency. The amplitude of the induced intensity modulation is directly proportional to the amount of slope in intensity; the steeper the intensity distribution, the greater the induced modulation. Therefore, if the intensity is converted to a n electronic signal through some radiation detector. and the modulated signal is detected and its amplitude measured, this measurement is proportional to the first derivative of intensity with respect to wavelength. Figure 3 presents the same description for a n intensity distribution containing curvature. In this case, the modulation of wavelength again induces intensity modulation, but now
A complete expansion introduces other derivative contributions to the amplitude of these waveforms, but these terms rapidly decrease and do not contribute extensively to the total value. In a complete theoretical description, it can be shown ( 5 )t h a t the amplitude of the nth harmonic is proportional to a weighted average of the nth derivative of intensity with respect to wavelength. The weighting function is maximum at A, and decreases to zero a t the range limits of wavelength modulation.
at t u i c e the modulation frequency. As the dip in intensity grows deeper (greater curvature or second derivative), the amplitude of the induced intensity modulation increases. The amplitude of the induced modulation a t twice the wavelength modulation frequency is now proportional to the second derivative of intensity with respect to wavelength. The exact mathematical theory supports what Figures 2 and 3 imply. By expanding the intensity distribution about a given wavelength A, in a Taylor series, the intensity can be represented as a function of wavelength and time by substituting A = A, d sin (wt). Here “ d ” is the a m plitude of wavelength modulation, and u is the modulation frequency. T h a t is,
Illustrating the Theory
+
where (A,) is the nth derivative of Zwith respect to wavelength evaluated a t A., Substituting ( A - A,) = d sin (wt) gives:
I(A,,t)
I(A”) + I”’(A,)d sin (wt) =
+
Expanding the powers of sine into sines and cosines of multiple angles yields: I(X,*t) =
d sin ( w t )
I(A,)
+
I(1l(A0)
x
+ -d?4I c 2 (A,)- dT 2I ’ ~ ’ ( A , ) X
cos ( 2 wt)
+...
(6) In this limited expression, the a m plitude of the sin (wt) and cos ( 2 wt) wave forms are proportional to
respectively, evaluated a t A,.
Figure 2. Optical measurement of slope
As shown previously, a second-derivative spectrometer can theoretically be applied to the measurement of gas concentration through Beer’s law. The instrument measures the amplitude of the waveform a t frequency 2 w coming from the detector. This amplitude is proportional to d2a/dAz if measured a t a wavelength where “a” is a maximum. Maximum sensitivity is achieved at wavelengths where the absorption coefficient is peaked, or where a gas exhibits narrow band absorption. Such is the case for nitric oxide, and this example can illustrate the improved sensitivity achieved by use of derivative spectroscopy over direct absorption spectroscopy. Nitric oxide has three narrow absorption bands in the ultraviolet. By use of direct spectroscopy, the intensity can be related directly to concentration a t low concentrations. When the product abc is much less than unity, Beer’s law can be expressed as:
I
=
1,(1 - abc) ( 5 )
Ioe-ohc
and 1
for some particular wavelength A,. Compare this expression to t h a t from a second-derivative spectrome-
ter. From Equation 6 the signal, S, is given as:
From Equation 3
dA2
=
(-bcd?a/dX’)I
(10
where A is chosen a t a maximum of a and d2Z0/dA2 is assumed negligible. Therefore,
From Figure 3. it is apparent that the magnitude of S is approximately that of ( I , - I). T h a t is, the amplitude of the 2 1c signal is about that of the depth of the dip in intensity. This is confirmed by choosing a realistic function for the absorption coefficient. Let “a” be a Gausian distribution -[(A,, - , \ i t I :. a - a,e
(1‘)
where z is the half width of the band a t a = a , / e . Now,
Substituting into Equation 11 yields:
If the modulation amplitude, cl. is chosen as 2 z, then
S
=
Ia,bc zz I , - I
(15)
The signal levels from the two techniques are, then, approximately the same. However, there is one important difference. The observable in the case of a derivative spectrometer is a signal from the detector which is proportional to concentration. Noise in this signal will limit detectability. The dominant noise is photomultiplier shot noise, which has a current value, i.,,,,, = G(2eiAf)’l2 (16)
Figure 3. Optical measurement of curvature ANALYTICAL CHEMISTRY, VOL. 45, NO. 13, NOVEMBER 1973
1133 A
Figure 4. Diagram of second-derivative g a s analyzer
where G is the PMT gain, e is the charge of en electron, i is the cathode current, and Lf is the frequency band width. In the case of nitric oxide, a typical absorbance limitation by use ofderivative techniques is (abc) = 5 x 10-5. In a direct spectrometer this same noise is present in the intensity signals. However. it is not the dominant factor in determining sensitivity. Two large signals, 1and I,, must be measured and then subtracted. The difference measurement. which is very small a t low absorbance. limits the detectable concentration. Even with double-beam compensating spectrophotometers, a n absorbance of 5 x lo-5cannot be approached. for this requires a measurement accuracy of I and 1, to 0.0057c . IVI easurement accuracy of concentration is proportional to total intensity in this case, whereas measurement accuracy in the derivative case is still proportional to concentration. instrumentation As indicated by the previous examples, derivative spectroscopy has proved to be a n extremely effective technique for trace gas analysis. Instrumentation has been developed 1134 A
ANALYTICAL CHEMISTRY, VOL
which is capable of specifically measuring concentrations into the low parts per billion range. The instrumentation is designed around second derivative spectroscopy, and the gases which are detected all exhibit ultraviolet or visible molecular absorption band spectra. The absorption bands must be less t h a n 40 A wide to achieve good sensitivity. A scanning second-derivative spectrometer coupled with a multipass absorption cell is presently marketed for trace gas analysis. An optical schematic of the instrument is shown in Figure 4. Radiation from an ultraviolet or visible source is spectrally dispersed by a grating monochomator so that the radiation leaving the exit slit is monochromatic in wavelength. This wavelength is varied in time by laterally oscillating the entrance slit posit ion. The entrance slit is electromechanically displaced sinusoidally a t a frequency of 45 Hz and a peak-to-peak displacement of to 1 mm. depending upon t h e desired amplitude of wavelength modulation. The wavelength entering the absorption cell is then centered about a value set by t h e grating position but is also modulated slightly in time at 45 Hz. If the 45, NO. 13, NOVEMBER 1973
slit oscillation amplitude is f j / 2 mm, the wavelength modulation is ~ 1 5 A . The absorption cell contains three spherical mirrors in a "White cell" arrangement (6).The diverging light is collected by one mirror at the far end of the 1-m long cell and focused upon the mirror at the near end. The light then diverges to the second mirror a t the far end which focuses it to a new position on the near end mirror. The process can be repeated u p to 32 times. but normal operation is a t 12 passes ofthe light through the sample in the cell. The light finally exits the absorption cell and is focused on a detector. a photomultiplier tube. The gas sample containing the particular compound to be measured is normally drawn continuously through the absorption cell. although it can be trapped inside the Teflon-lined cell. The output signal from the photomultiplier tube is electronically analyzed for a 90-Hz component. This component is generally small in a m plitude compared to the total dc output voltage and is buried in noise. However, a voltage is obtained from a coil fixed to the oscillating entrance slit and moving in a magnetic field which provides the exact frequency and phase of wavelength modulation. By electronically doubling the frequency of this signal. it is used as a reference to lock in frequency and phase with the 90-Hz signal within the photomultiplier tube output. By use of these standard phase-lock amplifier techniques. the amplitude of the 90-Hz signal is presented as a d c voltage. This is the signal proportional to the second derivative of' intensity with respect to wavelength. T o obtain a signal proportional to gas concentration. the derivative voltage signal is electronically divided by the total output voltage of the PM tube. The result is the desired voltage ratio. proportional to id")
ml1
As the grating position is slowly changed through gearing to motors. the center wavelength ofthe radiation passing through the sample slowly scans through the spectrum. The output of the instrument will differ from zero only at those wavelengths where curvature is present in the received radiation. If a gas is present which exhibits absorption bands. a peak similar in shape to Figure 1 will be recorded as the center wavelength scans across the absorption band. Since the output signal is linear with concentration at t h e extremums of this peak, as previously discussed, the peak height above a line drawn between peak minima is a n observable which is linear with concentration.
Figure 5. Derivative spectrum of ni-
tric oxide
A derivative spectrum of a gas sample can be analyzed for the different gases present by the peak locations, and each gas can be measured for quantity by the peak height of an associated peak. Figures 5 through 8 are second-derivative spectra of several gases. Note that no two spectra are alike. Each peak reflects the presence of an absorption band in the ultraviolet or visible spectral region, the energy transitions therefore being very large and resulting in new geometrical configurations of the molecule. Unlike infrared absorption bands with low energy transitions, the ultraviolet bands are relatively insensitive to temperature and pressure. The locations of the bands reflect the allowed energy transitions of the molecule and are therefore unique. Many gases can be detected and measured by this technique. Table I lists some of these and gives the lower limits of detection (based on a signalto-noise ratio of two) by using the described scanning instrumentation. Upper limits of detection are nearly unlimited, as the pathlength of radiation passing through the sample can be reduced to keep total light absorption from occurring. The technique offers extremely simple instrumentation for singlecomponent analysis. In this case, the derivative spectrometer is tuned to center at one wavelength rather than scan in wavelength. For example, a nitrogen dioxide analyzer is tuned to a center wavelength of 4480 A in the visible spectrum, the location of a strong absorption band. Tuning and wavelength modulation is accomplished by interference filters rather than a grating monochromator, which greatly simplifies the design. Because of the greater intensity 1136 A
Figure 6. Derivative spectrum of SUIfur dioxide
Figure 7. Derivative spectrum of nitrogen dioxide
throughput of filters as compared to a monochromator and because the derivative signal is continuously analyzed a t the NO2 peak, the sensitivity is improved so that measurements below 10 ppb are obtained reliably for nitrogen dioxide as compared to 40 ppb with a scanning instrument. This latter specific application of derivative spectroscopy stresses the value of the technique, for it accomplishes very simply a direct determination of nitrogen dioxide, a measurement which previously could not be made by any other technique. Other Applications of Derivative Spectroscopy
There are obvious applications for derivative spectroscopy beyond molecular absorption analysis. One specific application which has proved successful in the field of emission spectroscopy is the determination of trace nitrogen in high-purity argon. In this instrument, the sample passes through a quartz discharge tube where it is excited by a high voltage field. The emitted radiation, then, is split into two paths. One passes through an interference filter tuned to a nitrogen discharge line, the transmission wavelength of which is modulated &5 at 45 Hz. The second beam passes through a filter of constant wavelength transmission which is tuned t o a n argon line. The two beams are then combined and focused on a detector. The induced 90-Hz intensity modulation measures the intensity of the nitrogen line. The dc intensity measures the effective strength of the total discharge since argon concentration is constant. The ratio of these two voltages is therefore dependent only upon nitrogen concentration. S u b parts per million concentrations
ANALYTICAL CHEMISTRY, VOL. 45,
NO. 13,
NOVEMBER 1973
Figure 8. Derivative spectrum of ben-
zene
Table 1. Mlnimum Detectability of Some Compeunds hy U s e of a Second-Derivative Gas Analyzer Compound
Nitric oxide Nitrogen dioxide Sulfur dioxide ozone Ammonia Benzene Toiuem XYfOM
Styrene Formaidehyde Benzaidehyde ketaldehysle Mercury vapor
ppb
5
40 1 40 1 25 50
100 100 200
100 400 0.5
the Nicolet 1028A
a time averaging system for magnetic resonance spectrometers that gives you results like these: IS
The Nicolet 1020A is an easy-to-use time averaging system combining unique data processing capabilities with extremely useful display and readout routines. Using a normalized signal averaging method, the 1020A presents a true average on its built-in 5” CRT display. The complete spectrum (or any expanded portion of it) is continuously displayed throughout the averaging process. Thumbwheel switches select sweep times from 60 milliseconds to 1,999 seconds for a 2048-word sweep with 4-digit accuracy. The 1020A can provide sweep current to the spectrometer and can compensate for spectrometer field drift when using the Field Stabilization mode. The calibrated sweep widths are front panel selected and can be matched to any spectrometer. The analog-to-digital converter (ADC) resolution is 12 bits (one part in 4096) for sweep times over 1 second and 9 bits (one part in 512) for sweep times less than 1 second. Total memory capacity is 2048 words, 24 bits long. The upper 12 bits store the final average while the lower 12 may be used to store the integral. Averaged spectra may be stored in halves of memory and-added to or subtracted from one another. Digital integration is built-in and does not destroy the original spectrum thus allowing simultaneous display of spectrum and integral. Baseline correction can be accomplished during display of the integral to assure optimal results. Another unique display feature is the ability to expand the displayed spectrum in both the X and Y axes by factors of 2 up to a total expansion of 64 times. (Typical expansion examples are shown to the right). In addition to this analog display a simultaneous numerical display shows the amplitude and address values of any data point selected by a moveable electronic cursor. Write or call collect for comDlete details or to arrange a demonstration in your laboratory.
1 . Averaged spectrum of 2-butyl-1 P-dihydropyridene with spectrum expanded on amplitude scala by a factor of 2.
2 . integral of Figure 1 averaged spectrum.
3. Figure 1 spectrum and Figure 2 integral displayed simultaneously.
4 . Averaged spectrum expanded by a factor of 4 on both the amplitude scale and the frequency scale.
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In Europe Nicolet Instrument GmbH Goerdeler Strasse 48, D-605 Offenbach am Maln West Germany 061 1 /852028 Telex 841 /4185411
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Summary
In summary, the value of derivative spectroscopy can be recognized and applications chosen to benefit from the technique if the following fundamentals and properties are understood. Derivative spectroscopy senses changes in intensity with wavelength, such as slope and curvature. The output of a derivative spectrometer is expressible mathematically as a derivative of intensity with respect to wavelength. Therefore, a variable which. in a given application, is expressible as a function of the derivative of intensity can produce a predictable signal from a derivative spectrometer. The signal from a derivative spectrometer often relates to a parameter in a more usable fashion than the relation between that parameter and intensity. Sensitivity, linearity, and bipolarity are possible properties of derivative signals which may better serve the application. Finally, derivative spectroscopy can extract more information contained within a radiation intensity spectral distribution than is accessible through direct spectroscopy techniques because of the specific measurement of the derivatives of t h a t distribution. References ( I ) A . Griese and C . French. rlppl. S p e c t r m . . 9 , 78 (19551. ( 2 ) G. Collier and F. Singleton, ./.A p p l . C'hem.. 6,495 (19561. 13) G. Bonfialioli and P. Brooctlo, ,-tpp/. O p t 3 , I d 7 (1964) 141 F. Stauffer and H Sakai. [ b i d T. 61 (
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of nitrogen can be measured since the nitrogen line is very narrow and therefore creates large local curvature. Derivative spectroscopy has been used to extract very weak emission lines from a flame background (7) and improved detectability has been reported by use of derivative spectroscopy in atomic absorption work (8). Applications have been made successfully in fluorescence spectral analysis (9).Here, the ability to amplify small features in a distribution with derivative techniques has been used to resolve overlapping fluorescence spectra of several compounds.
e
1968)
(5i-R; Hager and R . Anderson, J . O p t . S O Cd. m e r . . 60, 1444 (1970). (6) R. If-hite. [bid..3 2 , 285 (194%). ( 7 ) LV. Snelleman. W . Rains. K . Yee. H. Cook. and 0. Menis. A n a / . C'hem.. 42, 394 (1970).
(8)R. Elser and J . IVinefordner. ibid..44,
698 (1972)). Skoki e / An ah e i in / B i r ni i n gham /Ci n c i n na ti /C1eve lan d/Da 11a s / (9) T. O ' H a i e r and B. Keppler. Paper I o . Denver/Detroit/Springfield,N.J ./ T o r o n t o / ~ l o n t r e a l / V ~ ~ ~ ~ o ~ ~ v e r 305. Pittsburgh Conference on Analyti-
cal Chemistry and Applied Spectroscopy. Cleveland, Ohio. 1972. CIRCLE 1 8 2
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