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
dA
d
f
λ
λ= ′( )
d A
d
f
2
2
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 zeroorder
spectra.
A first-order derivative is the rate
of change of absorbance with
respect to wavelength. A firstorder
derivative starts and finishes
at zero. It also passes through zero
at the same wavelength as lmax
of
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 lmax
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 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 1
Absorbance and derivative spectra of a Gaussian band
Obtaining derivative spectra
Derivative spectra can be obtained
by optical, electronic, or mathematical
methods. Optical and
electronic techniques were used
on early UV-Visible spectrophotometers
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 smoothing
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 electromechanical
device. The first and
second derivatives may be generated
using this technique. It is
popular for dedicated spectrophotometer
designs used in, for
example, environmental monitoring.
First-derivative spectra may
also be generated by a dual
wavelength spectrophotometer.
The derivative spectrum is generated
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
nth
order
l wavelength
A absorbance
e extinction coefficient
b sample path length
c sample concentration
For single component quantification
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 evenorder
derivatives there is a peak
maximum or minimum at the same
lmax
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-tonoise
ratio but may lead to increased
sensitivity to interference
from other components.
the amplitude and wavelength
shift of the derivatives varies with
scan speed, slit width, and resistance-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. Firstderivative
spectra may be calculated
simply by taking the difference
in absorbance between two
closely spaced wavelengths for all
wavelengths :
Where the derivative amplitude,
Dl
, is calculated for a wavelength
intermediate between the two
absorbance wavelengths.
For the second-derivative determination
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:
A a a al
l
λ λ λ= + + +0 1 Κ
D
A A
λ λ
λ λ λ
λ+
+
=
−
∆
∆
∆/
( )
2
dA
d S
dA
dtλ
=
1
A b c= ε
dA
d
d
d
bc
λ
ε
λ
=
d A
d
d
d
bc
n
n
n
n
λ
ε
λ
=
D
A A A
λ
λ λ λ λ λ
λ
=
− +− +( )∆ ∆
∆
2
2
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 coefficients
a0
...al
at each wavelength
multiplied by the factorial of the
order are the derivative values: a1
is the first derivative, 2xa2
the
second derivative, 6xa3
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 algorithm
in most commercial instruments.
Other techniques for
calculating derivatives, for example,
using Fourier Transforms,
are available but are not commercially
popular.
One consequence of these mathematical
methods for the calculation
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 decreased
by the loss of lowerorder
data such as constant offset
factors.
For example, the absorbance
spectrum of the steroid testosterone
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 evenorder
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 lmax
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
Absorbance
Features and applications
For clarity the points made in the
following discussion are illustrated
using computer-generated examples.
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.
A a a an
n
= + +0 1
1
λ 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 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 lmax
of the component
bands.
Although the bands have been
resolved there is no indication of
whether these arise from two
chromophores in a single compound
or in two different compounds.
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 improvements
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 repositioning)
effects. Because the first
derivative of a constant absorbance
offset is zero, using the firstderivative
spectra always eliminates
such baseline shifts and
improves the accuracy of quantification.
This is illustrated in figure
3 where a 0.1 A offset, that would
cause a 10 % quantitative error for
the analyte, is completely eliminated
by the first derivative.
Figure 3
Background elimination
D
W
n
n=
1
D
D
W
W
X
n
Y
n
Y
n
X
n
=
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, Dn
, of a Gaussian band
in the nth
derivative is inversely
proportional to the original
bandwidth, W, raised to the nth
degree:
Thus for two coincident bands of
equal intensity but different
bandwidth in the zero order, the
nth
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 component
and to reduce error caused
by scattering.
Scattering is a common problem in
biological analyses resulting from
the measurement of small particulates
present in the sample.
Scattering is inversely proportional
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 absorbance,
using derivatives discriminates
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 contribution.
Quantification at this wavelength
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 quantification
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 identification
of trace levels of a component
in the presence of a strongly
absorbing matrix. This is illustrated
in Figure 6. A trace, 0.01 A,
of a 40 nm NBW component with
lmax
at 500 nm was added to a
synthetic matrix. The matrix
comprised offset, second and
fourth-order scatter, and 320 nm
NBW components with lmax
at 300
and 600 nm. In absorbance mode
the presence of this component is
virtually undetectable. In secondorder
derivative mode its presence
is obvious.
1.0
0.5
0.0
500 600
0.01
0.0
-0.01
700400300
500 600 700400300
Figure 5
Scatter elimination
Figure 6
Matrix 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 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, quantification
of the analyte in the absorbance
mode without any correction
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/l2
).
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 absorbance
mode, quantification of
caffeine is inaccurate because of
the matrix effect but good accuracy
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 considerations
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 spectroscopy
but wavelength reproducibility
and signal-to-noise are of
increased importance.
The increased resolution of
derivative spectra puts increased
demands on the wavelength
reproducibility of the spectrophotometer.
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 spectrum
as standard, may often be
used to achieve the same analytical
goals but without the reduced
signal-to-noise penalty.
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Copyright © 1995 Agilent Technologies
All Rights Reserved. Reproduction, adaptation
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