Instrumentation
Derivative Spectroscopy with Emphasis on Trace Gas Analysis Robert N. Hager, Jr. LSI/Spectrometrics, Inc. One Inverness Dr. Englewood, Colo. 80110
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 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 (;) as a means for resolving two spectral lines of very nearly equal wavelength. To accomplish this resolution, the first or second derivatives of an 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 and their derivatives ANALYTICAL CHEMISTRY, VOL. 45, NO. 13, NOVEMBER 1973 · 1131 A
specific one is lacking. If the first de rivative is measured at this wave length also, information is added. A constant first derivative (slope) indi cates that the absorption varies lin early 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 at this wavelength which greatly restricts the possible candi date gas. So, in this example, deriva tive spectroscopy supplies informa tion 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, I0, il lustrates the intensity before passing through an absorbing medium; the solid is the distribution after some ra diation is absorbed. Note that the amount of light absorbed at λ 0 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 at λ„ alone was monitored, no discrimination could be made between the two com pounds. However, as the first and second derivatives are evaluated at λ0, 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 the broad band absorber produces no signal. It is true that intensity alone can be measured as a function of wave length by using a direct spectrometer. The information on slope and curva ture is then within the output data but not readily accessible. It is not an "observable." Intensity is the observ able, the signal from the instrument, and this must be processed again to gain information on slope or curva ture. This processing adds inaccura cies 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 observ able can be related directly to desired information, then accuracy and sensi tivity can often be substantially im proved. Theory of Derivative Absorption Spectroscopy The preceding example of radia tion absorption was not chosen at random, for much work has been done in the application of derivative spectroscopy to the analysis of trace
gases. To look a bit deeper into this application, consider the relation be tween the derivatives of intensity and gas concentration. Starting with the function that contains the informa tion, the intensity distribution is pre dicted by Beer's law: A = In y
= abc
where A is absorbance, a is the ab sorption coefficient as a function of wavelength for a particular gas, b is the pathlength of radiation through the gas, and c is the gas concentra tion. The derivatives of intensity can be influenced by a(X) and Ι0(λ), al though the latter can be selected somewhat to fit a desired situation. That is, the radiation source can be chosen to produce only minor contri butions in the derivative signal with in a working wavelength region. Then, the first derivative dl d\
dl0 = d\
( abc) da , T . ^ Oc/» exp (-ofcc)
Source of Derivative Signal
exp
(1)
or d/
1 dl„
da
dx'1 = ijx-
bc
dx
^
The two terms in the above equa tion 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 deriva tive spectrometer, a signal can be ob tained which is linear with concentra tion. This is more desirable than the logarithmic relationship by use of di rect spectroscopy. Secondly, the sensitivity to concen tration of the signal depends now on a physical property, the rate of change in absorption coefficient at a particular wavelength, da/d\. This fact may or may not be useful in a given application, but it certainly in troduces a new special class of com pounds that would be applicable to the analysis technique. The same mathematics can be car ried a step farther to obtain the sec ond derivative of intensity with re spect to wavelength. The result is a bit more complex and produces four terms as follows: d 21
' , τ - l d2l° + (h daX + bc dvJI - h "dxr \ d\) 2 dlo da bc
h^Xd\
~
bc
da2
dK
(3)
The first term is a constant mea suring the curvature in the radiation source. The second and third terms
1132 A · 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 oc curs. At this point, curvature will be a maximum. Also, at 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 concentra tion is obtained from the last term. Now, the sensitivity factor is the cur vature in absorption coefficient which further discriminates those com pounds which will create a secondderivative signal.
A true derivative spectrometer ob tains the derivative signal optically, not electronically. This is extremely important, for an intensity signal which is conditioned electronically, in fact, takes the derivatives with re spect to time. The wavelength is var ied linearly with time so that the re sult looks like a derivative of intensi ty with respect to wavelength. How ever, time fluctuations in the intensi ty signal (noise) will be sensed, and the resulting derivative of the noise will be noisier. If a technique senses only changes in intensity with wave length, time fluctuations will not be sensed and noise is minimized. The optical technique used in a de rivative spectrometer is wavelength modulation. The wavelength of the radiation passing through the sample is modulated sinusoidally with time, the wavelength amplitude of modula tion generally being the width of the spectral feature to be analyzed. Consider Figure 2. An intensity dis tribution is shown which contains slope (first derivative). If radiation is modulated about λ 0 in time, a modu lation in intensity will be induced. This induced modulation frequency is equal to the wavelength modulation frequency. The amplitude of the in duced intensity modulation is direct ly proportional to the amount of slope in intensity; the steeper the intensity distribution, the greater the induced modulation. Therefore, if the intensi ty is converted to an electronic signal through some radiation detector, and the modulated signal is detected and its amplitude measured, this mea surement is proportional to the first derivative of intensity with respect to wavelength. Figure 3 presents the same descrip tion for an intensity distribution con taining curvature. In this case, the modulation of wavelength again in duces intensity modulation, but now
at twice the modulation frequency. As the dip in intensity grows deeper (greater curvature or second deriva tive), the amplitude of the induced intensity modulation increases. The amplitude of the induced modulation at twice the wavelength modulation frequency is now proportional to the second derivative of intensity with re spect to wavelength. The exact mathematical theory supports what Figures 2 and 3 imply. By expanding the intensity distribu tion about a given wavelength λ 0 in a Taylor series, the intensity can be represented as a function of wave length and time by substituting λ = λ„ + d sin (wt). Here "d" is the am plitude of wavelength modulation, and w is the modulation frequency. That is,
/(λ) = 7(λ0) + 7(λ0)(λ - λ„) + 7^(λ„) (λ ~
λ )2
°
+
...
(4)
where 7"11 (λ 0 ) is the nth derivative of I with respect to wavelength evalu ated at λ 0 . Substituting (λ — λ 0 ) = d sin (wt) gives: I(Xmt)
= I(X„) In)(\„)d 2
, d Ι ,2) (λ„)
+
sin (wt)
+
2
sin (wt) 2!
-·
(5)
Expanding the powers of sine into sines and cosines of multiple angles yields: KKt)
= 7(λ„) +
dsin(wt)
7«>(A0)x
+ ~P2>(\„) 4
^/'2l(X„)x 4
cos (2 wt) + ... (6) In this limited expression, the am plitude of the sin (wt) and cos (2 wt) wave forms are proportional to 2 dl Λ d I and dX dX2 respectively, evaluated at λ„.
Figure 2. Optical measurement of slope
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 descrip tion, it can be shown (5) that the am plitude of the nth harmonic is pro portional to a weighted average of the nth derivative of intensity with re spect to wavelength. The weighting function is maximum at λ0 and de creases to zero at the range limits of wavelength modulation. Illustrating the Theory
As shown previously, a second-de rivative spectrometer can theoretical ly be applied to the measurement of gas concentration through Beer's law. The instrument measures the ampli tude of the waveform at frequency 2 w coming from the detector. This amplitude is proportional to d2a/d\2 if measured at a wavelength where " a " is a maximum. Maximum sensi tivity is achieved at wavelengths where the absorption coefficient is peaked, or where a gas exhibits nar row band absorption. Such is the case for nitric oxide, and this example can illustrate the improved sensitivity achieved by use of derivative spec troscopy over direct absorption spec troscopy. Nitric oxide has three narrow ab sorption bands in the ultraviolet. By use of direct spectroscopy, the inten sity can be related directly to concen tration at low concentrations. When the product abc is much less than unity, Beer's law can be expressed as:
ter. From Equation 6 the signal, S, is given as: d2 s
dlI\,
d\ix ' λ»
= τ
(9
>
From Equation 3 •jj-, = (-bcd2a/dX2)I
(10)
where λ is chosen at a maximum of α and d2I0/d\2 is assumed negligible. Therefore, d2a
-d-
(11)
s-^r-fcdX^
From Figure 3, it is apparent that the magnitude of S is approximately that of (I0 - I). That is, the ampli tude of the 2 w signal is about that of the depth of the dip in intensity. This is confirmed by choosing a realistic function for the absorption coeffi cient. Let " a " be a Gausian distribution —
—['λ,, — X)l s /
, .,
a — a„e (12) where ζ is the half width of the band at a = a0/e. Now, d2a
—cio
Μ
= -τ
at λ = λ„
(13)
Substituting into Equation 11 yields: "
(d2abc) T S = -T1r-/ 0
(M)
If the modulation amplitude, d, is chosen as 2 z, then S = Iajbc
^
I„ -
I
(15)
The signal levels from the two techniques are, then, approximately the same. However, there is one im portant difference. The observable in I = I„e-''
