Uses of Derivative Spectroscopy

Application Note

UV-Visible Spectroscopy Anthony J. Owen

Derivative spectroscopy uses first or higher derivatives of absorbance

with respect to wavelength for qualitative analysis and for

quantification. The concept of derivatizing spectral data was first

introduced in the 1950s, when it was shown to have many advantages.

However, the technique received little attention primarily because of the

complexity of generating derivative spectra using early UV-Visible

spectrophotometers. The introduction of microcomputers in the late

1970s made it generally practicable to use mathematical methods to

generate derivative spectra quickly, easily and reproducibly. This

significantly increased the use of the derivative technique.

In this application note we review briefly the mathematics and

generation methods of derivative spectroscopy. We illustrate the

features and applications using computer-generated examples.

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A f= ( )λ

dAd

fλ

λ= ′( )

d A

df

2

2λλ= ′′( )

Figure 1 shows a computersimulation of the effects ofderivatization on the appearanceof a simple Gaussian absorbanceband. Derivative spectra arealways more complex than zero-order spectra.

A first-order derivative is the rateof change of absorbance withrespect to wavelength. A first-order derivative starts and finishesat zero. It also passes through zeroat the same wavelength as �

max of

the absorbance band. Either sideof this point are positive andnegative bands with maximum andminimum at the same wavelengthsas the inflection points in theabsorbance band. This bipolarfunction is characteristic of allodd-order derivatives.

The most characteristic feature ofa second-order derivative is anegative band with minimum atthe same wavelength as themaximum on the zero-order band.It also shows two additionalpositive satellite bands either sideof the main band. A fourth-orderderivative shows a positive band.

A strong negative or positive bandwith minimum or maximum at thesame wavelength as �max of theabsorbance band is characteristicof the even-order derivatives.

Note that the number of bandsobserved is equal to the derivativeorder plus one.

Introduction

If a spectrum is expressed asabsorbance, A, as a function ofwavelength,��, the derivativespectra are:

Zero order

First order

Second order

0.0

0.5

1.0

300 400 600 700500 300 400 600 700500

0.0

0.5

1.0Absorbance Absorbance

0.01

0.00

-0.01

300 400 600 700500 300 400 600 700500

2.0E-04

-6.0E-04

0.0E+00

-2.0E-04

-4.0E-04

300 400 600 700500 300 400 600 700500

0.0E+00

-2.0E-04

-1.0E-04

1.0E-04

2.0E-04

0.0E+00

-5.0E-07

5.0E-07

1.0E+063rd derivative 4rd derivative

2nd derivative1st derivative

Figure 1Absorbance and derivative spectra of a Gaussian band

Obtaining derivative spectra

Derivative spectra can be obtainedby optical, electronic, or math-ematical methods. Optical andelectronic techniques were usedon early UV-Visible spectropho-tometers but have largely beensuperseded by mathematicaltechniques. The advantages of themathematical techniques are thatderivative spectra may be easilycalculated and recalculated withdifferent parameters, and smooth-ing techniques may be used toimprove the signal-to-noise ratio.

Optical and electronictechniquesThe main optical technique iswavelength modulation, where thewavelength of incident light israpidly modulated over a narrowwavelength range by an electrome-chanical device. The first andsecond derivatives may be gener-ated using this technique. It ispopular for dedicated spectropho-tometer designs used in, forexample, environmental monitor-ing. First-derivative spectra mayalso be generated by a dualwavelength spectrophotometer.The derivative spectrum is gener-ated by scanning with eachmonochromator separated by asmall constant wavelength differ-ence.

First and higher-order derivativescan be generated using analogresistance capacitance devices.These generate the derivative as afunction of time as the spectrum isscanned at constant speed(dA/dt=S). For the first derivative:

Higher-order derivatives areobtained by using successivederivators. The electronic methodsuffers from the disadvantage that

Quantification

If we assume that the zero-orderspectrum obeys Beer’s law, thereis a similar linear relationshipbetween concentration andamplitude for all orders of deriva-tive:

Zero order

First order

nth order

� wavelengthA absorbance� extinction coefficientb sample path lengthc sample concentration

For single component quantifica-tion the selection of wavelengthsfor derivative spectra is not assimple as for absorbance spectrabecause there are both positiveand negative peaks. For the even-order derivatives there is a peakmaximum or minimum at the same�

max as the absorbance spectrum

but for the odd-order derivativesthis wavelength is a zero crossing

point.

Taking the difference between thehighest maximum and the lowestminimum gives the best signal-to-noise ratio but may lead to in-creased sensitivity to interferencefrom other components.

the amplitude and wavelengthshift of the derivatives varies withscan speed, slit width, and resis-tance-capacitance gain factor.

Mathematical techniquesTo use mathematical techniquesthe spectrum is first digitized witha sampling interval of ��. The sizeof �� depends on the naturalbandwidth (NBW) of the bandsbeing processed and of thebandwidth of the instrument usedto generate the data. Typically, forUV-Visible spectra, the NBW is inthe range 10 to 50 nm. First-derivative spectra may be calcu-lated simply by taking the differ-ence in absorbance between twoclosely spaced wavelengths for allwavelengths :

Where the derivative amplitude,D�, is calculated for a wavelengthintermediate between the twoabsorbance wavelengths.

For the second-derivative determi-nation three closely-spacedwavelength values are used:

Higher-order derivatives can becalculated from similar expres-sions.

This method involves simple linearinterpolation between adjacentwavelengths. A better method isthat proposed by Savitzky andGolay. To calculate the derivativeat a particular wavelength, �, awindow of ±n data points isselected and a polynomial is fittedusing the least squares method:

A a a all

λ λ λ= + + +0 1 Κ

DA A

λ λλ λ λ

λ++=

−∆

∆

∆/

( )2

dA

d S

dA

dtλ= 1

A b c= ε

dA

d

d

dbc

λελ

=

d A

d

d

dbc

n

n

n

nλε

λ=

DA A A

λλ λ λ λ λ

λ=

− +− +( )∆ ∆

∆2

2

An advantage of this method isthat it can be used to smooth thedata. If the polynomial order, l, isless than the number of datapoints (2n+1) in the window, thepolynomial generally cannot gothrough all data points and thusthe least squares fit gives asmoothed approximation to theoriginal data points. This featurecan be used to counteract thedegradation of signal-to-noise thatis inherent in the derivatizationprocess (see below). The coeffi-cients a

0...a

l at each wavelength

multiplied by the factorial of theorder are the derivative values: a

1

is the first derivative, 2xa2 the

second derivative, 6xa3 the third

derivative, and so on.

Savitzky and Golay developed avery efficient method to performthe calculations and this is thebasis of the derivatization algo-rithm in most commercial instru-ments. Other techniques forcalculating derivatives, for ex-ample, using Fourier Transforms,are available but are not commer-cially popular.

One consequence of these math-ematical methods for the calcula-tion of derivatives is that datapoints at the beginning and end ofthe wavelength range are lost. Ifthree data points are used for theprocess then one data point will belost at each end of the range foreach derivative order. If five pointsare used then two points will belost and so on.

It should be clearly understoodthat, although transformation of aUV-Visible spectrum to its first orhigher derivative usually yields amore complex profile than thezero-order spectrum (see below),the intrinsic information content isnot increased. In fact, it is de-creased by the loss of lower-order data such as constant offset

factors.

For example, the absorbancespectrum of the steroid testoster-one has a single, broad, featurelessband centered at about 330 nm butthe second derivative has six quitedistinctive peaks.

ResolutionAs figure 1 shows, the derivativecentroid bandwidth of the even-order derivatives decreases withincreasing order. Relative to thezero-order spectrum the derivativecentroid bandwidth for a Gaussianband is observed to decrease to53 %, 41 %, and 34 % of the originalbandwidth in the second, fourth,and sixth orders respectively.

This feature may be used inqualitative analysis to identify thepresence of two analytes with verysimilar��

max values that are not

resolved in the absorbancespectrum. Figure 2 shows acomputer simulation.

1.5

1.0

0.5

0.0

400 500 600

5.0E-06

400 500 600

0.0E-06

-5.0E-06

Absorbance

Absorbance

Features and applications

For clarity the points made in thefollowing discussion are illustratedusing computer-generated ex-amples. In the figures, dotted linesshow the baseline, dashed linesshow component spectra, and solidlines show the analyte spectrummade up from the componentspectra.

GraphicsAs shown in figure 1 there is anincrease in the number of bands ashigher orders of derivative arecalculated. This increase in thecomplexity of the derivative spectracan be very useful in qualitativeanalysis, either for characterizingmaterials or for identification.Spectra that are very similar inabsorbance mode may revealsignificant differences in thederivative mode.

Figure 2Resolution enhancement

Other background effects that aredirectly proportional to higherorders of wavelength with thegeneral form:

can be eliminated by using higherorders of derivative but suchspectral features with exactly thisform are very uncommon so thiseffect has little practical use.

A a a ann= + +0 1

1λ λ. . .

0.0

0.5

1.0

300 400 600 700500 300 400 600 700500

0.0

0.5

1.0

In absorbance mode, when twoGaussian bands with 40 nm NBWand separated by 30 nm, are addedthe result is a single band with amaximum midway between thetwo component bands. The twocomponents are not resolved. Inthe fourth derivative the presenceof these two bands is clearlyvisible with maxima centeredclose to the �

max of the component

bands.

Although the bands have beenresolved there is no indication ofwhether these arise from twochromophores in a single com-pound or in two different com-pounds. It is often claimed thatthis increased resolution and theincreased differentiation betweenspectra in the derivative modeallows multicomponent analysisof mixtures of components withsimilar spectra that cannot beresolved in the absorbance mode.

However, as noted above theinformation content of derivativespectra is, in fact, less than theabsorbance spectra and it caneasily be shown that the improve-ments in quantitative accuracy arethe result of other effects asdescribed below.

Background eliminationA common, unwanted effect inspectroscopy is baseline shift. Thismay arise either from instrument(lamp or detector instabilities) orsample handling (cuvette reposi-tioning) effects. Because the firstderivative of a constant absor-bance offset is zero, using the first-derivative spectra always elimi-nates such baseline shifts andimproves the accuracy of quantifi-cation. This is illustrated in figure3 where a 0.1 A offset, that wouldcause a 10 % quantitative error forthe analyte, is completely elimi-nated by the first derivative.

Figure 3Background elimination

DW

nn= 1

D

D

W

WXn

Yn

Yn

Xn

=

DiscriminationProbably the most importanteffect of the derivative process isthat broad bands are suppressedrelative to sharp bands and thissuppression increases withincreasing derivative order.

This arises from the fact that theamplitude, Dn, of a Gaussian bandin the nth derivative is inverselyproportional to the originalbandwidth, W, raised to the nth

degree:

Thus for two coincident bands ofequal intensity but differentbandwidth in the zero order, thenth derivative amplitude of thesharper band, X, is greater thanthat of the broader band, Y, by afactor that is dependent on therelative bandwidth and thederivative order:

Figure 4 shows the effect of takingderivatives of two bands, one with160 nm NBW and one with 40 nmNBW. In absorbance mode theyhave equal amplitude, in firstderivative the narrower band hasfour times greater amplitude andin the second derivative it hassixteen times the amplitude.

This property is used to improvethe accuracy of quantification of anarrow band component in thepresence of a broad band compo-nent and to reduce error causedby scattering.

Scattering is a common problem inbiological analyses resulting fromthe measurement of small particu-lates present in the sample.

Scattering is inversely propor-tional to the fourth (Rayleigh,small particles) or second(Tyndall, larger particles) power ofthe wavelength. Because therelationship is inverse, the use ofderivatives will not eliminate thescattering component from thespectrum as has been claimed insome publications. However,because the scattering componentresembles a very broad absor-bance, using derivatives discrimi-nates against it and reduces itseffect on quantification.

1.0

0.5

0.0

400 500 600

400 500 600

400 500 600

0.05

0.0

-0.05

0.005

0.000

-0.010

-0.005

Absorbance

2nd derivative

1st derivative

Figure 4Discrimination against broad bands

Figure 5 shows an absorbanceband with 40 nm NBW and thesame band in the presence of ascattering background. Withoutany correction, the amplitude at500 nm is 1.0920 A instead of 1.0 Abecause of the scattering contribu-tion. Quantification at this wave-length results in an error of 9.2 %.Using the first derivative thecontribution from the scatteringcomponent is reduced such that,using peak maximum to minimum,the signal in the presence ofscattering is 0.02992 instead of0.03024, that is a quantificationerror of only -1.1 %. This ability todiscriminate against scatteringcomponents is widely used in theanalysis of biological fluids thatcontain a high level of particulates,and in pharmaceutical analyseswhere particulate excipients intablets and capsules cause quanti-fication errors.

Matrix suppressionThe analytical problem is often notsimply scattering, baseline shift, orunwanted broad absorbingcomponents. It is a combination oftwo or more of these that resultsin a broad absorbing backgroundmatrix.

In qualitative analyses,derivatization often allows thedetection and positive identifica-tion of trace levels of a componentin the presence of a stronglyabsorbing matrix. This is illus-trated in Figure 6. A trace, 0.01 A,of a 40 nm NBW component with�

max at 500 nm was added to a

synthetic matrix. The matrixcomprised offset, second andfourth-order scatter, and 320 nmNBW components with �

max at 300

and 600 nm. In absorbance modethe presence of this component isvirtually undetectable. In second-order derivative mode its presenceis obvious.

1.0

0.5

0.0

500 600

0.01

0.0

-0.01

700400300

500 600 700400300

Figure 5Scatter elimination

Figure 6Matrix supression

1.0

0.5

0.0

6.0E-05

2.0E-05

-2.0E-05

600 800400200

Absorbance

Absorbance

4.0E-05

0.0E+00

Anthony J. Owen is product

manager at Agilent Technologies,

Waldbronn, Germany.

For the latest information and services visit ourworld wide web site:http://www.agilent.com/chem

In quantitative analyses,derivatization improves theaccuracy of quantification in thepresence of interference caused bya broad absorbing component,matrix, or scattering. Thus in theexample given above, quantifica-tion of the analyte in the absor-bance mode without any correc-tion results in an error of nearly5000 % (absorbance of 0.502 Ainstead of 0.01 A). Using thebaseline-to-valley signal of thesecond-order derivative the erroris -2.1% (2.37 x 10-5 instead of2.42 x 10-5 A/�2).

An example of discriminationagainst a broad absorbing matrixis the quantification of caffeine insoft drinks. Soft drinks generallycontain a mixture of natural andsynthetic products with addedcolorants, resulting in a broadfeatureless absorbance over awide wavelength range. In absor-bance mode, quantification ofcaffeine is inaccurate because ofthe matrix effect but good accu-racy can often be achieved usingthe second-derivative spectra.

Instrument considerations

Virtually all current UV-Visiblespectrophotometers generatederivative spectra by mathematicalmeans so instrument consider-ations for generation of derivativespectra by optical and electronictechniques are not discussed.Instrument requirements forderivative spectroscopy are, ingeneral, similar to those forconventional absorbance spectros-copy but wavelength reproducibil-ity and signal-to-noise are ofincreased importance.

The increased resolution ofderivative spectra puts increaseddemands on the wavelengthreproducibility of the spectropho-tometer. Small wavelength errorscan result in much larger signalerrors in the derivative mode thanin the absorbance mode.

The negative effect ofderivatization on signal-to-noisealso puts increased demands onlow noise characteristics of thespectrophotometer. It is anadvantage in this case, if thespectrophotometer can scan andaverage multiple spectra beforederivatization to improve furtherthe signal-to-noise ratio.

For the derivatization process it isimportant to be able to control thedegree of smoothing that isapplied in order to adapt todiffering analytical problems. Inthe case of the Savitzky-Golaymethod this means being able tovary the order of polynomial andthe number of data points used.

Signal-to-noise ratio

An unwanted effect of thederivatization process is that thesignal-to-noise ratio decreases ashigher orders of derivatives areused. This follows from thediscrimination effect and the factthat noise always contains thesharpest features in the spectrum.Thus, if the spectral data used inthe derivative calculation is at2 nm intervals, the noise has a2 nm bandwidth. If the analyteband has a bandwidth of 20 nmthen the signal-to-noise ratio of thefirst derivative is ten times worsethan the zero-order spectrum. Thedecrease in signal-to-noise ratiocan be reduced by using thesmoothing properties of theSavitzky-Golay polynomialsmoothing technique but greatcare must be taken as too high adegree of smoothing distorts thederivative spectrum.

Alternative techniques, such asusing a reference wavelength orfull spectrum multicomponentanalysis with a scattering spec-trum as standard, may often beused to achieve the same analyti-cal goals but without the reducedsignal-to-noise penalty.

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