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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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Af= ()λ

λ=

′

()

λ=

′′

()

Figure 1 shows a computer

simulation of the effects of

derivatization on the appearance

of a simple Gaussian absorbance

band. Derivative spectra are

always more complex than zero-

order spectra.

A first-order derivative is the rate

of change of absorbance with

respect to wavelength. A first-

order derivative starts and finishes

at zero. It also passes through zero

at the same wavelength as l

max

the absorbance band. Either side

of this point are positive and

negative bands with maximum and

minimum at the same wavelengths

as the inflection points in the

absorbance band. This bipolar

function is characteristic of all

odd-order derivatives.

The most characteristic feature of

a second-order derivative is a

negative band with minimum at

the same wavelength as the

maximum on the zero-order band.

It also shows two additional

positive satellite bands either side

of the main band. A fourth-order

derivative shows a positive band.

A strong negative or positive band

with minimum or maximum at the

same wavelength as l

max

of the

absorbance band is characteristic

of the even-order derivatives.

Note that the number of bands

observed is equal to the derivative

order plus one.

Introduction

If a spectrum is expressed as

absorbance, A, as a function of

wavelength, l, the derivative

spectra are:

Zero order

First order

Second order

0.0

0.5

1.0

300

400

600 700

500

300

400

600 700

500

0.0

0.5

1.0

Absorbance

0.01

0.00

-0.01

300

400

600 700

500

300

400

600 700

500

2.0E-04

-6.0E-04

0.0E+00

-2.0E-04

-4.0E-04

300

400

600 700

500

300

400

600 700

500

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+06

3rd derivative 4rd derivative

2nd derivative

1st derivative

Figure 1

Absorbance and derivative spectra of a Gaussian band

Obtaining derivative spectra

Derivative spectra can be obtained

by optical, electronic, or math-

ematical methods. Optical and

electronic techniques were used

on early UV-Visible spectropho-

tometers but have largely been

superseded by mathematical

techniques. The advantages of the

mathematical techniques are that

derivative spectra may be easily

calculated and recalculated with

different parameters, and smooth-

ing techniques may be used to

improve the signal-to-noise ratio.

Optical and electronic

techniques

The main optical technique is

wavelength modulation, where the

wavelength of incident light is

rapidly modulated over a narrow

wavelength range by an electrome-

chanical device. The first and

second derivatives may be gener-

ated using this technique. It is

popular for dedicated spectropho-

tometer designs used in, for

example, environmental monitor-

ing. First-derivative spectra may

also be generated by a dual

wavelength spectrophotometer.

The derivative spectrum is gener-

ated by scanning with each

monochromator separated by a

small constant wavelength differ-

ence.

First and higher-order derivatives

can be generated using analog

resistance capacitance devices.

These generate the derivative as a

function of time as the spectrum is

scanned at constant speed

(dA/dt=S). For the first derivative:

Higher-order derivatives are

obtained by using successive

derivators. The electronic method

suffers from the disadvantage that

Quantification

If we assume that the zero-order

spectrum obeys Beer’s law, there

is a similar linear relationship

between concentration and

amplitude for all orders of deriva-

tive:

Zero order

First order

order

l wavelength

absorbance

e extinction coefficient

sample path length

sample concentration

For single component quantifica-

tion the selection of wavelengths

for derivative spectra is not as

simple as for absorbance spectra

because there are both positive

and negative peaks. For the even-

order derivatives there is a peak

maximum or minimum at the same

max

as the absorbance spectrum

but for the odd-order derivatives

this wavelength is a zero crossing

point.

Taking the difference between the

highest maximum and the lowest

minimum gives the best signal-to-

noise ratio but may lead to in-

creased sensitivity to interference

from other components.

the amplitude and wavelength

shift of the derivatives varies with

scan speed, slit width, and resis-

tance-capacitance gain factor.

Mathematical techniques

To use mathematical techniques

the spectrum is first digitized with

a sampling interval of Dl. The size

of Dl depends on the natural

bandwidth (NBW) of the bands

being processed and of the

bandwidth of the instrument used

to generate the data. Typically, for

UV-Visible spectra, the NBW is in

the range 10 to 50 nm. First-

derivative spectra may be calcu-

lated simply by taking the differ-

ence in absorbance between two

closely spaced wavelengths for all

wavelengths :

Where the derivative amplitude,

, is calculated for a wavelength

intermediate between the two

absorbance wavelengths.

For the second-derivative determi-

nation three closely-spaced

wavelength values are used:

Higher-order derivatives can be

calculated from similar expres-

sions.

This method involves simple linear

interpolation between adjacent

wavelengths. A better method is

that proposed by Savitzky and

Golay. To calculate the derivative

at a particular wavelength, l, a

window of ±n data points is

selected and a polynomial is fitted

using the least squares method:

Aaa a

λλ=+ ++

λλ

λλ λ

−

∆

()

Abc=ε

AAA

λλ λ λ λ

−+

( )

∆∆

∆

An advantage of this method is

that it can be used to smooth the

data. If the polynomial order, l, is

less than the number of data

points (2n+1) in the window, the

polynomial generally cannot go

through all data points and thus

the least squares fit gives a

smoothed approximation to the

original data points. This feature

can be used to counteract the

degradation of signal-to-noise that

is inherent in the derivatization

process (see below). The coeffi-

cients a

...a

at each wavelength

multiplied by the factorial of the

order are the derivative values: a

is the first derivative, 2xa

the

second derivative, 6xa

the third

derivative, and so on.

Savitzky and Golay developed a

very efficient method to perform

the calculations and this is the

basis of the derivatization algo-

rithm in most commercial instru-

ments. Other techniques for

calculating 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 data

points at the beginning and end of

the wavelength range are lost. If

three data points are used for the

process then one data point will be

lost at each end of the range for

each derivative order. If five points

are used then two points will be

lost and so on.

It should be clearly understood

that, although transformation of a

UV-Visible spectrum to its first or

higher derivative usually yields a

more complex profile than the

zero-order spectrum (see below),

the intrinsic information content is

not increased. In fact, it is de-

creased by the loss of lower-

order data such as constant offset

factors.

For example, the absorbance

spectrum of the steroid testoster-

one has a single, broad, featureless

band centered at about 330 nm but

the second derivative has six quite

distinctive peaks.

Resolution

As figure 1 shows, the derivative

centroid bandwidth of the even-

order derivatives decreases with

increasing order. Relative to the

zero-order spectrum the derivative

centroid bandwidth for a Gaussian

band is observed to decrease to

53 %, 41 %, and 34 % of the original

bandwidth in the second, fourth,

and sixth orders respectively.

This feature may be used in

qualitative analysis to identify the

presence of two analytes with very

similar l

max

values that are not

resolved in the absorbance

spectrum. Figure 2 shows a

computer 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

Features and applications

For clarity the points made in the

following discussion are illustrated

using computer-generated ex-

amples. In the figures, dotted lines

show the baseline, dashed lines

show component spectra, and solid

lines show the analyte spectrum

made up from the component

spectra.

Graphics

As shown in figure 1 there is an

increase in the number of bands as

higher orders of derivative are

calculated. This increase in the

complexity of the derivative spectra

can be very useful in qualitative

analysis, either for characterizing

materials or for identification.

Spectra that are very similar in

absorbance mode may reveal

significant differences in the

derivative mode.

Figure 2

Resolution enhancement

Other background effects that are

directly proportional to higher

orders of wavelength with the

general form:

can be eliminated by using higher

orders of derivative but such

spectral features with exactly this

form are very uncommon so this

effect has little practical use.

Aa a a

=+ +

λλ...

0.0

0.5

1.0

300

400

600

700

500

300

400

600

700

500

0.0

0.5

1.0

In absorbance mode, when two

Gaussian bands with 40 nm NBW

and separated by 30 nm, are added

the result is a single band with a

maximum midway between the

two component bands. The two

components are not resolved. In

the fourth derivative the presence

of these two bands is clearly

visible with maxima centered

close to the l

max

of the component

bands.

Although the bands have been

resolved there is no indication of

whether these arise from two

chromophores in a single com-

pound or in two different com-

pounds. It is often claimed that

this increased resolution and the

increased differentiation between

spectra in the derivative mode

allows multicomponent analysis

of mixtures of components with

similar spectra that cannot be

resolved in the absorbance mode.

However, as noted above the

information content of derivative

spectra is, in fact, less than the

absorbance spectra and it can

easily be shown that the improve-

ments in quantitative accuracy are

the result of other effects as

described below.

Background elimination

A common, unwanted effect in

spectroscopy is baseline shift. This

may arise either from instrument

(lamp or detector instabilities) or

sample handling (cuvette reposi-

tioning) effects. Because the first

derivative of a constant absor-

bance offset is zero, using the first-

derivative spectra always elimi-

nates such baseline shifts and

improves the accuracy of quantifi-

cation. This is illustrated in figure

3 where a 0.1 A offset, that would

cause a 10 % quantitative error for

the analyte, is completely elimi-

nated by the first derivative.

Figure 3

Background elimination

Discrimination

Probably the most important

effect of the derivative process is

that broad bands are suppressed

relative to sharp bands and this

suppression increases with

increasing derivative order.

This arises from the fact that the

amplitude, D

, of a Gaussian band

in the n

derivative is inversely

proportional to the original

bandwidth, W, raised to the n

degree:

Thus for two coincident bands of

equal intensity but different

bandwidth in the zero order, the

derivative amplitude of the

sharper band, X, is greater than

that of the broader band, Y, by a

factor that is dependent on the

relative bandwidth and the

derivative order:

Figure 4 shows the effect of taking

derivatives of two bands, one with

160 nm NBW and one with 40 nm

NBW. In absorbance mode they

have equal amplitude, in first

derivative the narrower band has

four times greater amplitude and

in the second derivative it has

sixteen times the amplitude.

This property is used to improve

the accuracy of quantification of a

narrow band component in the

presence of a broad band compo-

nent and to reduce error caused

by scattering.

Scattering is a common problem in

biological analyses resulting from

the 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 of

the wavelength. Because the

relationship is inverse, the use of

derivatives will not eliminate the

scattering component from the

spectrum as has been claimed in

some publications. However,

because the scattering component

resembles a very broad absor-

bance, using derivatives discrimi-

nates against it and reduces its

effect 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 4

Discrimination against broad bands

Figure 5 shows an absorbance

band with 40 nm NBW and the

same band in the presence of a

scattering background. Without

any correction, the amplitude at

500 nm is 1.0920 A instead of 1.0 A

because of the scattering contribu-

tion. Quantification at this wave-

length results in an error of 9.2 %.

Using the first derivative the

contribution from the scattering

component is reduced such that,

using peak maximum to minimum,

the signal in the presence of

scattering is 0.02992 instead of

0.03024, that is a quantification

error of only -1.1 %. This ability to

discriminate against scattering

components is widely used in the

analysis of biological fluids that

contain a high level of particulates,

and in pharmaceutical analyses

where particulate excipients in

tablets and capsules cause quanti-

fication errors.

Matrix suppression

The analytical problem is often not

simply scattering, baseline shift, or

unwanted broad absorbing

components. It is a combination of

two or more of these that results

in a broad absorbing background

matrix.

In qualitative analyses,

derivatization often allows the

detection and positive identifica-

tion of trace levels of a component

in the presence of a strongly

absorbing 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 matrix

comprised offset, second and

fourth-order scatter, and 320 nm

NBW components with l

max

at 300

and 600 nm. In absorbance mode

the presence of this component is

virtually undetectable. In second-

order derivative mode its presence

is obvious.

1.0

0.5

0.0

500 600

0.01

0.0

-0.01

700

400

300

500 600 700

400

300

Figure 5

Scatter elimination

Figure 6

Matrix supression

1.0

0.5

0.0

6.0E-05

2.0E-05

-2.0E-05

600 800

400

200

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 our

world wide web site:

http://www.agilent.com/chem

In quantitative analyses,

derivatization improves the

accuracy of quantification in the

presence of interference caused by

a broad absorbing component,

matrix, or scattering. Thus in the

example given above, quantifica-

tion of the analyte in the absor-

bance mode without any correc-

tion results in an error of nearly

5000 % (absorbance of 0.502 A

instead of 0.01 A). Using the

baseline-to-valley signal of the

second-order derivative the error

is -2.1% (2.37 x 10

-5

instead of

2.42 x 10

-5

A/l

An example of discrimination

against a broad absorbing matrix

is the quantification of caffeine in

soft drinks. Soft drinks generally

contain a mixture of natural and

synthetic products with added

colorants, resulting in a broad

featureless absorbance over a

wide wavelength range. In absor-

bance mode, quantification of

caffeine is inaccurate because of

the matrix effect but good accu-

racy can often be achieved using

the second-derivative spectra.

Instrument considerations

Virtually all current UV-Visible

spectrophotometers generate

derivative spectra by mathematical

means so instrument consider-

ations for generation of derivative

spectra by optical and electronic

techniques are not discussed.

Instrument requirements for

derivative spectroscopy are, in

general, similar to those for

conventional absorbance spectros-

copy but wavelength reproducibil-

ity and signal-to-noise are of

increased importance.

The increased resolution of

derivative spectra puts increased

demands on the wavelength

reproducibility of the spectropho-

tometer. Small wavelength errors

can result in much larger signal

errors in the derivative mode than

in the absorbance mode.

The negative effect of

derivatization on signal-to-noise

also puts increased demands on

low noise characteristics of the

spectrophotometer. It is an

advantage in this case, if the

spectrophotometer can scan and

average multiple spectra before

derivatization to improve further

the signal-to-noise ratio.

For the derivatization process it is

important to be able to control the

degree of smoothing that is

applied in order to adapt to

differing analytical problems. In

the case of the Savitzky-Golay

method this means being able to

vary the order of polynomial and

the number of data points used.

Signal-to-noise ratio

An unwanted effect of the

derivatization process is that the

signal-to-noise ratio decreases as

higher orders of derivatives are

used. This follows from the

discrimination effect and the fact

that noise always contains the

sharpest features in the spectrum.

Thus, if the spectral data used in

the derivative calculation is at

2 nm intervals, the noise has a

2 nm bandwidth. If the analyte

band has a bandwidth of 20 nm

then the signal-to-noise ratio of the

first derivative is ten times worse

than the zero-order spectrum. The

decrease in signal-to-noise ratio

can be reduced by using the

smoothing properties of the

Savitzky-Golay polynomial

smoothing technique but great

care must be taken as too high a

degree of smoothing distorts the

derivative spectrum.

Alternative techniques, such as

using a reference wavelength or

full spectrum multicomponent

analysis with a scattering spec-

trum as standard, may often be

used to achieve the same analyti-

cal goals but without the reduced

signal-to-noise penalty.

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