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**Introduction to FT-IR spectroscopy**

FOURIER TRANSFORMATION MEASUREMENT EVALUATION

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Introduction

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**Friedrich Wilhelm Herschel**

Discovery of infrared light In the year 1800 the astronomer Friedrich Wilhelm Herschel analyzed the spectrum of sunlight. Herschel created the spectrum by directing sunlight through a glass prism so that the light was divided into its different colors. He measured the heating ability of each color using thermometers with blackened bulbs. When he measured the temperature just beyond the red part of the spectrum he noticed some kind of invisible radiation. Much to his surprise he found that the area close to the red part (i.e. an area apparently devoid of sunlight) had the highest heating ability of all. Herschel concluded that there must be a different kind of light beyond the red portion of the spectrum, which is not visible to the human eye. This kind of light became known as “infrared” (below red) light. Herschel then placed a water-filled container between the prism and thermometer and observed that the temperature measured was lower than the one measured without the water. Consequently, the water must partially absorb the radiation. In addition, Herschel could prove that depending on how the prism was rotated (i.e. depending on the spectral range) the difference in the temperatures measured for each color varied. This was the beginning of infrared spectroscopy. Infrared spectroscopy measures the infrared light that is absorbed by a substance. This absorption depends on the wavelength of the light. Friedrich Wilhelm Herschel ( )

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**Sinusoidal wave of wavelength **

The electromagnetic spectrum Visible light and infrared light are two types of electromagnetic radiation, but with different wavelengths, or frequencies. In general, electromagnetic radiation is defined by the wavelength or the linear frequency . The wavelength is the distance between two maxima on a sinusoidal wave. The frequency is the number of wavelengths per unit time. Since all electromagnetic waves travel at the speed of light, the frequency corresponding to a given wavelength can be calculated as: = c/ According to the Plank’s Radiation Law, the frequency of electromagnetic radiation is proportional to its energy. E = h• In infrared spectroscopy wavenumber is used to describe the electromagnetic radiation. Wavenumber is the number of wavelengths per unit distance. For a wavelength in microns, the wavenumber, , in cm-1, is given by = 10000/ ~ Sinusoidal wave of wavelength ~

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**The electromagnetic spectrum**

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**Interaction of radiation and matter**

If matter is exposed to electromagnetic radiation, e.g. infrared light, the radiation can be absorbed, transmitted, reflected, scattered or undergo photoluminescence. Photoluminescence is a term used to designate a number of effects, including fluorescence, phosphorescence, and Raman scattering. Matter Photoluminescence Incident light beam Absorption Transmission Reflection Scattering

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**Mechanical model of a vibrating diatomic molecule**

Interaction of radiation and matter Vibration theory IR spectroscopy is based on the absorption of infrared light by the substance to be measured. This absorption excites molecular vibrations and rotations, which have frequencies that are the same as those within the infrared range of the electromagnetic spectrum. The following simple model of an harmonic oscillator used in classical physics describes IR absorption. If atoms are considered to be particles with a given mass, then the vibrations in a diatomic molecule (e.g. HCl) can be described as follows: Mechanical model of a vibrating diatomic molecule

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Vibration theory The molecule consists of mass m1 and m2 connected by a spring. At equilibrium, the distance between the two masses is r0. If the molecule is stretched by an amount r = x1 + x2, then a restoring force, F, is produced. If the spring is released, the system will vibrate around the equilibrium position. According to Hooke’s Law, for small deflections the restoring force is proportional to the deflection: F = -k . r Since the force acts in a direction opposite to the deflection the proportionality constant, or force constant, k, is negative in sign. The force constant is called the spring constant in the mechanical model, whereas in a molecule the force constant is a measure of the bond strength between the atoms. For a harmonic oscillator it is possible to calculate the vibrational frequency, , of a diatomic molecule as follows: being the reduced mass.

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Vibration theory On the basis of the equation above it is possible to state the following: 1) The higher the force constant k, i.e. the bond strength, the higher the vibrational frequency, , (in wavenumbers). ~ 3 absorption peaks for different force constants. Note that by convention, in infrared spectroscopy wavenumbers are plotted right-to-left; i.e. highest wavenumber to the left.

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Vibration theory 2) The larger the vibrating atomic mass, the lower the vibrational frequency, , (in wavenumbers). ~ 3 absorption peaks for different atomic masses. Note that by convention, in infrared spectroscopy wavenumbers are plotted right-to-left; i.e. highest wavenumber to the left.

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**Potential energy curve for a harmonic oscillator**

Vibration theory For the harmonic oscillator model, the potential energy well is symmetric. According to quantum-mechanical principles molecular vibrations can only occur at discrete, equally spaced, vibrational levels, where the energy of the vibration is given by: Ev=(v + ½) h v = 0, 1, 2, 3, ... Where h is Plank’s constant and v is the vibrational quantum number. Even in case of v = 0, which is defined as the ground vibrational level, a molecule does vibrate: Ev= ½ h When absorption occurs, the molecule acquires a clearly defined amount of energy, (E = h ), from the radiation and moves up to the next vibrational level (v = +1). If the molecule moves down to the next vibrational level (v = -1), a certain amount of energy is emitted in the form of radiation. This is called emission. For a harmonic oscillator, the only transitions permitted by quantum mechanics are up or down to the next vibrational level (v = 1). Potential energy curve for a harmonic oscillator

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**Potential energy curve for an anharmonic oscillator**

Vibration theory A more accurate model of a molecule is given by the anharmonic oscillator. The potential energy is then calculated by the Morse equation, and is asymmetric. The energy levels are no longer equally spaced, and are given by: Ev=(v + ½) h - (v + ½)2 xGl h where xGl is the anharmonicity constant. The anharmonic oscillator model allows for two important effects: 1) As two atoms approach each other, the repulsion will increase very rapidly. 2) If a sufficiently large vibrational energy is reached the molecule will dissociate (break apart). This is called the dissociation energy. In the case of the anharmonic oscillator, the vibrational transitions no longer only obey the selection rule v = 1. This type of vibrational transition is called fundamental vibration. Vibrational transitions with v = 1, 2, 3, ... are also possible, and are termed overtones. Potential energy curve for an anharmonic oscillator

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**Vibration theory What kind of molecules absorb infrared light?**

Infrared light can only be absorbed by a molecule if the dipole moment of the specific group of atoms changes during the vibration. The greater the change in dipole moment, the stronger the corresponding IR absorption band will be. Heteronuclear diatomic molecule Vibrations not accompanied by changes in the dipole moment can not be excited by absorption of IR light, and are termed IR inactive. A consequence of this is that homonuclear diatomic molecules, e.g. H2 or O2, do not have any IR spectrum. Homonuclear diatomic molecule Note: Raman scattering occurs if the polarizability of the of the bond changes during the vibration. This means that IR-inactive vibrations are Raman active if the polarizability changes. Raman and IR spectra therefore complement each other.

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**NIR MIR FIR Separation of spectral ranges 15,000 cm-1 4,000 cm-1**

The mid-infrared, or MIR, is the spectral range from 4,000 to 400 cm-1 wavenumbers. In this range fundamental vibrations are typically excited. In contrast, in the ‘near-infrared’, or NIR, spectral range, which covers the range from 12,500 to 4,000 cm-1 wavenumbers, overtones and combination vibrations are excited. The far infrared’, or FIR, spectral range is between 400 and about 5 cm-1 wavenumbers. This range covers the vibrational frequencies of both backbone vibrations of large molecules, as well as fundamental vibrations of molecules that include heavy atoms (e.g. inorganic or organometallic compounds). NIR MIR FIR 15,000 cm-1 4,000 cm-1 400 cm-1 5 cm-1

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**Fourier Transformation**

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**Fourier Transformation**

Since the development of the first spectrophotometers in the beginning of 20th century a rapid technological development has taken place. The first-generation spectrometers were all dispersive. Initially, the dispersive elements were prisms, and later on they changed over to gratings. In the mid 1960s IR spectroscopy witnessed a revival due to the advent of spectrometers that utilized the Fourier transform (FT-IR). These second-generation spectrometers, with an integrated Michelson interferometer, provided some significant advantages compared to dispersive spectrometers. Today, almost every spectrometer used in mid-infrared spectroscopy is is of the Fourier transform type. This is the reason why only FT-IR technology will be described in the following. Bruker Optics has specialized in the field of FT-IR spectroscopy since 1974, and is one of the leading manufacturers of FT-IR, FT-NIR and FT-Raman spectrometers throughout the world. The spectrometers are developed for analytical chemistry, life science, process, and many other fields.

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**General FT-IR spectrometer layout**

The working principle of an FT-IR spectrometer Infrared light emitted from a source (e.g. a SiC glower) is directed into an interferometer, which modulates the light. After the interferometer the light passes through the sample compartment (and also the sample) and is then focused onto the detector. The signal measured by the detector is called the interferogram. General FT-IR spectrometer layout

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**Michelson interferometer**

Fixed mirror M1 The interferometer is the heart of an FT-IR spectrometer. The collimated light from the infrared source impinges on a beamsplitter, which ideally transmits 50% of the light and reflects the remaining part. Having traveled the distance L the reflected light is hits a fixed mirror M1, where it is reflected and hits the beamsplitter again after a total path length of 2L. The transmitted part of the beam is directed to a movable mirror M2. As this mirror moves back and forth around L by a distance Dx, the total path length is 2(L + Dx). The light returning from the two mirrors is recombined at the beamsplitter, with the two beams having a difference in path length of 2Dx. The beams are spatially coherent and interfere with each other when recombined. Movable mirror M2 L x Source Beamsplitter L + x x=0 Detector Michelson interferometer

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**Monochromatic source detector signal**

Origin of the interferogram The upper figure on the right shows the interferogram generated by the detector for a monochromatic source. The interferometer splits and recombines the two beams with a relative phase difference that depends on the mirror displacement, or optical retardation. The two beams undergo constructive interference, yielding a maximum detector signal, if the optical retardation is an integral multiple of the wavelength λ, i.e. if 2 Dx = n λ (n = 0, 1, 2, ...). Destructive interference, and a minimum detector signal, occur if 2Dx is a multiple of λ/2. The complete functional relationship between I(x) and x is given by the cosine function I(x) = S(ν) cos (2 π x) In which we use wavenumber, = 1/λ, which is more common in FTIR spectroscopy. S( ) is the intensity of a monochromatic spectral line at wavenumber , as shown in the lower figure on the right. Intensity shown as a function of frequency is called a spectrum, and can be obtained by Fourier transformation of the signal that is a function of optical retardation. The cosinusoidal interference pattern from a monochromatic source is very useful, because it enables a very precise tracking of the movable mirror. All state-of-the-art FT-IR spectrometers use the interference pattern of the monochromatic light emitted by an HeNe laser to monitor the mirror position. The IR interferogram is digitized exactly at the zero crossings of the laser interferogram. Detector signal Intensity Optical Retardation Monochromatic source detector signal ~ Spectrum ~ ~ ~ Intensity Frequency Monochromatic source

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**consisting of 9 single frequencies**

Origin of the interferogram Optical retardation Intensity Nine wavelengths Since spectrometers are equipped with a polychromatic light source (i.e. many wavelengths) the interference already mentioned occurs at each wavelength, as shown in the upper figure on the right. The interference patterns produced by each wavelength are summed to get the resulting interferogram, as shown in the second figure. At the zero path difference of the moving mirror (Dx=0) both paths all wavelengths have a phase difference of zero, and therefore undergo constructive interference. The intensity is therefore a maximum value. As the optical retardation increases, each wavelength undergoes constructive and destructive interference at different mirror positions. The third figure shows the intensity as a function of frequency (I.e. the spectrum), and we now have nine lines. Optical retardation Intensity Resulting detector signal: Spectrum consisting of 9 single frequencies Intensity Frequency

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**Frequency distribution of a black body source**

Origin of the interferogram Spectrometers are equipped with a broadband light source, which yields a continuous, infinite number, of wavelengths, as shown in the figure on the left. The interferogram is the continuous sum, i.e. the integral, of all the interference patterns produced by each wavelength. This results in the intensity curve as function of the optical retardation shown in the second figure. At the zero path difference of the interferometer (Dx=0) all wavelengths undergo constructive interference and sum to a maximum signal. As the optical retardation increases different wavelengths undergo constructive and destructive interference at different points, and the intensity therefore changes with retardation. For a broadband source, however, all the interference patterns will never simultaneously be in phase except at the point of zero path difference, and the maximum signal occurs only at this point. This maximum in the signal is referred to as the “centerburst” IR-source Resulting detector signal Intensity Intensity Optical retardation Frequency Frequency distribution of a black body source Resulting interferogram (detector signal after modulation by a Michelson interferometer)

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**Advantages of FTIR spectroscopy IR spectrometer principle**

1) The sampling interval of the interferogram, dx, is the distance between zero-crossings of the HeNe laser interferogram, and is therefore precisely determined by the laser wavelength. Since the point spacing in the resulting spectrum, d , is inversely proportional to dx, FT-IR spectrometers have an intrinsically highly precise wavenumber scale (typically a few hundredths of a wavenumber). This advantage of FT spectrometers is known as CONNES’ advantage. 2) The JAQUINOT advantage arises from the fact that the circular apertures used in FTIR spectrometers has a larger area than the slits used in grating spectrometers, thus enabling higher throughput of radiation. 3) In grating spectrometers the spectrum S(ν) is measured directly by recording the intensity at successive, narrow, wavelength ranges. In FT-IR spectrometers all wavelengths from the IR source impinge simultaneously on the detector. This leads to the multiplex, or FELLGETT’S, advantage. The combination of the Jaquinot and Fellgett advantages means that the signal-to-noise ratio of an FT spectrometer can be more than 10 times that of a dispersive spectrometer. ~ Dispersive IR spectrometer FT-IR spectrometer

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The Fourier Transform Data acquisition results in a digitized interferogram, I(x), which is converted into a spectrum by means of the mathematical operation called a Fourier Transform (FT). The general equation for the Fourier Transform is applicable to a continuous signal. If the signal (interferogram) is digitized, however, and consists of N discrete, equidistant points, then the discrete version of the FT (DFT) must be used: S(k . Δ ) = Σ I(n Δx) exp (i2πk n/N) The continuous variables x and have been replaced with n Dx and k D , representing the n discrete interferogram points and the k discrete spectrum points. The fact that we now have a discrete, rather than continuous, function, and that it is only calculated for a limited range of n (i.e. the measured interferogram has a finite length) leads to important effects known as the picket-fence effect and leakage. ~ ~ ~

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Zero filling 1,796 1,798 1,800 1,802 1,804 1,806 1,808 Wavenumber, cm-1 0.35 0.40 0.45 0.50 0.55 Single channel The picket-fence effect occurs if the interferogram contains frequency components which do not exactly coincide with the data point positions, k.Δ , in the spectrum. The effect can be thought of as viewing the spectrum through a picket fence, thereby hiding those frequencies that are behind the pickets, i.e. between the data point positions k.Δ . In the worst case, if a frequency component exactly between two sampling positions, a signal reduction of 36% can occur. The picket-fence effect can be reduced by adding zeros to the end of the interferogram (zero filling) before the DFT is performed. This interpolates the spectrum, increasing the number of points per wavenumber. The increased number of frequency sampling positions reduces the error caused by the picket-fence effect. Generally, the original interferogram size should always be at least doubled by zero filling, i.e. zero filling factor (ZFF) of two is chosen. Zero-filling interpolates using the instrument line-shape, and in most cases is therefore superior to polynominal or spline interpolation methods that are applied in the spectral domain. ~ ~ Zero-filling factor 2 1,796 1,798 1,800 1,802 1,804 1,806 1,808 Wavenumber, cm-1 0.35 0.40 0.45 0.50 0.55 Single channel Zero-filling factor 8

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**BOXCAR (no apodization) 3-TERM BLACKMAN-HARRIS**

In a real measurement, the interferogram can only be measured for a finite distance of mirror travel. The resulting interferogram can be thought of as an infinite length interferogram multiplied by a boxcar function that is equal to 1 in the range of measurement and 0 elsewhere. This sudden truncation of the interferogram leads to a sinc( ) (i.e. sin( )/ ) instrumental lineshape. For an infinitely narrow spectral line, the peak shape is shown at the top of the figure on the right. The oscillations around the base of the peak are referred to as “ringing”, or “leakage”. The solution to the leakage problem is to truncate the interferogram less abruptly. This can be achieved by multiplying the interferogram by a function that is 1 at the centerburst and close to 0 at the end of the interferogram. This is called apodization, and the simplest such function is a ramp, or “triangular apodization”. The choice of a particular apodization function depends on the objectives of the measurement. If the maximum resolution of 0.61/L is required, then boxcar apodization (i.e no apodization) is used. If a resolution loss of 50% (compared to the maximum resolution of 0.61/L) can be tolerated, the HAPP-GENZEL or, even better, 3-Term BLACKMAN-HARRIS function is recommended. ~ ~ ~ B Triangular C Trapezoidal D HAPP-GENZEL E 3-TERM BLACKMAN-HARRIS

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Phase correction To this point, only the ideal case has been considered, in which the point of zero path difference is the same for all wavelengths. In practice, due to both optical and electronic effects, this is not the case, and the sinusoidal interference patterns for different wavelengths are slightly shifted with respect to each other. These phase shifts, or “phase errors” lead to asymmetry in the interferogram. This asymmetry is corrected during the DFT using one of a number of “phase correction” algorithms. The algorithm generally used was developed by Larry Mertz, and is therefore called “Mertz phase correction”.

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**The spectral resolution**

d If a spectrum consists of a pair of narrow spectral lines a distance d apart, then the interferogram exhibits a periodic beat pattern that is repeated after at multiples of the optical path difference of 1/d. The smaller the spacing between the spectral lines, the greater the period of this beat pattern. For an optical retardation of L, the “nominal resolution” is therefore given by 1/L. The actual, measured, resolution is also affected by optical considerations (most importantly aperture size) and apodization. In the case of triangular apodization, the instrument line shape is a sinc( )2 function, and two lines with centers separated by 1/L will have a dip of 20% between them. This is the Rayleigh criterion for the resolution of two lines. Wavenumber, cm-1 ~

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**Transmission spectrum**

To calculate a transmission spectrum the following steps need to be performed: An interferogram measured without any sample in the optical path is Fourier transformed. This results in the single-channel reference spectrum R(). Detector signal Optical retardation Fourier transformation 0.40 0.30 Single-channel intensity 0.20 0.10 4,000 3,500 3,000 2,500 2,000 1,500 1,000 500 Wavenumber, cm-1

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**Transmission spectrum**

To calculate a transmission spectrum the following steps need to be performed: An interferogram measured without any sample in the optical path is Fourier transformed. This results in the single-channel reference spectrum R(). A second interferogram, measured with the sample in the optical path, is Fourier transformed. This results in the single-channel sample spectrum S(). S() looks similar to the reference spectrum, but shows less intensity at those wavenumbers where the sample absorbs radiation. Detector signal Optical retardation Fourier transformation 0.40 0.30 Single-channel intensity 0.20 0.10 4,000 3,500 3,000 2,500 2,000 1,500 1,000 500 Wavenumber, cm-1

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**Transmission spectrum**

500 1,000 1,500 2,000 2,500 3,000 3,500 4,000 Wavenumber, cm-1 0.10 0.20 0.30 0.40 Single-channel intensity 40 60 80 100 Transmittance [%] 20 Division To calculate the transmission spectrum the following steps need to be performed: an interferogram measured without any sample in the optical path is Fourier transformed. This results in the so-called single-channel reference spectrum (). A second interferogram, measured with the sample in the optical path, is Fourier transformed. This results in the single-channel sample spectrum S(). S() looks similar to the reference spectrum, but shows less intensity at those wavenumbers where the sample absorbs radiation. The final transmission spectrum T() is obtained by dividing the sample spectrum by the reference spectrum: T() = S()/R()

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IR Measurements

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**IR Measurements Sample preparation**

There are many measurement methods available for the analysis of samples using IR spectroscopy. It is this multitude of different measurement techniques which provides the necessary flexibility to analyze many different types of samples. The method selected for analysis usually can and should be optimized for the given sample. This short guide, however, cannot explain all measurement methods in detail. This guide will instead give an overview of the important factors affecting IR sample analysis and will focus on the two most common techniques: transmission and ATR (attenuated total reflection). Sample preparation The necessity for and nature of sample preparation depends on the composition, physical attributes, and absorption properties of the sample. Additionally, the spectral range of interest and the measurement method applied will determine how much and what type of preparation will be needed. In this manuscript you will find details on the sample preparation necessary for IR transmission analysis and basic information on ATR spectroscopy. Generally speaking, transmission analysis requires some preparation of the sample to ensure that the sample is optically thin. ATR spectroscopy usually requires little to no sample preparation.

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**Measuring spectra in transmission**

This section describes several common procedures for sample preparation prior to collecting IR spectra in transmission. It is important to select the best preparation method for a given sample. Some samples may require trial and error to get an acceptable spectrum. Irrespective of the physical sample condition, the material should be as homogeneous as possible. Variations in concentration or chemical compositions within the sample can lead to confusing or erroneous results. Chemical composition The position of absorption peaks and their respective intensities depend on the intrinsic chemical structure of the “unknown”. The characteristic absorption features of the sample are an important factor when selecting a suitable method to prepare the sample. To obtain meaningful spectra, strongly absorbing samples must either be very thin or diluted with a solvent or powder which are poorly absorbing. Significant prior knowledge of the sample may be necessary to predict whether strongly absorbing bands will be evident. Reference spectra of the sample or components of the sample can prove helpful.

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**Concentration Physical properties**

The intensity of a peak in an absorption spectrum is directly proportional to the concentration of the sample substance concerned, as defined by the Lambert-Beer Law: A = • b • c A: Absorption maximum at a given wavelength : Molar absorption coefficient (absorption probability at a given wavelength) b: Sample pathlength (for samples in a cell) or sample thickness (pressed pellets for films) c: Sample concentration If ε of an absorption band is high, the concentration of the unknown (c) needs to be lower to get a peak (A) with an acceptable intensity. If the peak (A) is weak, the unknown may need to have a higher concentration (increase c) or the sample has to be analyzed with a higher layer thickness (increase b). Physical properties Special cells are required to analyze gas, liquids or solvents (unless a free-standing liquid film on transparent substrate material is used). Powders must be fine, of uniform grain size and as finely-ground as possible. Very smooth, free-standing films may exhibit interference patterns (also known as fringing), which may cause fine spectra details to be concealed.

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**Sample preparation to measure spectra in transmission**

A detailed description of all procedures necessary to prepare samples used for analysis would be beyond the scope of this guide. This section describes some methods normally used to prepare samples and provides further information on how to select the right preparation method. Please note that the most well suited preparation method (for a particular sample) may only become evident after acquiring spectra using more that one technique. If one method turns out to be inappropriate, select a second method you think might be appropriate or ask your applications expert. The following sample preparation methods are most common: 1.) No sample preparation (example: free-standing film) 2.) Apply a thin film between 2 thin, transparent support plates (NaCl, KBr etc.). Typically, a few droplets are applied between two thin transparent support plates (sandwich). The sandwich is fixed in the sample holder. The spacing between the two support plates is very small (typically < 0.01mm), but sufficient for most samples. If the solution is very volatile, it may need to be filled in a sealed cell using very thin spacers. Proceed with care to prevent the solution from dissolving or affecting the support plates (e.g. mounting an aqueous solution with two NaCl support plates would not be a good choice - the plates would dissolve). Thin film between support plates

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**Sample preparation to measure spectra in transmission**

A detailed description of all procedures necessary to prepare samples used for analysis would be beyond the scope of this guide. This section describes some methods normally used to prepare samples and provide further information on how to select the right preparation method. Please note that the most suited preparation method (for a particular sample) may only become evident after acquiring spectra using more that one technique. If one method turns out to be inappropriate, select a second method you think might be appropriate or ask your applications expert. The following methods are the most common to prepare samples: 1.) No sample preparation (example: free-standing film) 2.) Apply a thin film between 2 thin, transparent support plates (NaCl, KBr etc.). 3.) Dissolve a sample powder in a spectrophotometrically pure solvent and - apply the solution on a transparent support plate or - fill it into an IR liquid cell It is very important to use solvents of high quality which are spectrophotometrically pure to avoid spurious bands associated with contaminates present in the solvent. The solvent selected should not be highly absorbing within the spectral range of interest. If the solvent within the particular spectral range is highly absorbing, the bands of the unknown may be concealed by solvent peaks and can thus be overlooked. For the absorption properties of different solvents, see the reference literature. After solvating the sample, evaporate the solvent onto a transparent support plate (salt window), yielding a thin sample film. The window can then be fixed in a sample holder for analysis. Alternatively, you can fill the solution into a liquid cell for analysis. A reference spectrum (background) of the cell with the pure solvent should be used. The typical cell capacity is between 0.1 and 1 ml. Microcells with lower capacity are also available. Dissolve solid sample in a solvent

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**Sample preparation to measure spectra in transmission**

A detailed description of all procedures necessary to prepare samples used for analysis would be beyond the scope of this guide. This section describes some methods normally used to prepare samples and provide further information on how to select the right preparation method. Please note that the most suited preparation method (for a particular sample) may only become evident after acquiring spectra using more that one technique. If one method turns out to be inappropriate, select a second method you think might be appropriate or ask your applications expert. The following methods are the most common to prepare samples: 1.) No sample preparation (example: free-standing film) 2.) Apply a thin film between 2 thin, transparent support plates (NaCl, KBr etc.). 3.) Dissolve a sample powder in a spectrophotometrically pure solvent and - apply the solution on a transparent support plate or - fill it into an IR liquid cell. 4.) Grind a powder sample into fine particles using a mortar and pestle and prepare a nujol suspension. This method is useful if the sample can be ground to very fine particles. The sample is finely ground and mixed with a liquid in which the sample is not soluble. The resulting suspension is smeared onto an IR transparent support plate and analyzed. Nujol, a kind of paraffin oil, is frequently used to prepare suspensions. Prepare a suspension

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**Sample preparation to measure spectra in transmission**

A detailed description of all procedures necessary to prepare samples used for analysis would be beyond the scope of this guide. This section describes some methods normally used to prepare samples and provide further information on how to select the right preparation method. Please note that the most suited preparation method (for a particular sample) may only become evident after acquiring spectra using more that one technique. If one method turns out to be inappropriate, select a second method you think might be appropriate or ask your applications expert. The following methods are the most common to prepare samples : 1.) No sample preparation (example: free-standing film) 2.) Apply a thin film between 2 thin, transparent support plates (NaCl, KBr etc.). 3.) Dissolve a sample powder in a spectrophotometrically pure solvent and - apply the solution on a transparent support plate or - fill it into an IR liquid cell. 4.) Grind a powder sample into fine particles using a mortar and pestle and prepare a nujol suspension. 5.) For optically thin solid samples, grind the sample into fine particles and press the powder to obrain a very fine and even pellet. 6.) Grind a solid sample into fine particles and mix it with a IR transparent powder (KBr etc.). Press the powder mixture to a very thin pellet. A solid sample is ground. The fine particles are mixed with a supporting medium (e.g. NaCl, KBr) and pressed into a pellet. The sample should be dilute with respect to the medium (1-5%). Press a pellet

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**Sample preparation to measure spectra in transmission**

A detailed description of all procedures necessary to prepare samples used for analysis would be beyond the scope of this guide. This section describes some methods normally used to prepare samples and provide further information on how to select the right preparation method. Please note that the most suited preparation method (for a particular sample) may only become evident after acquiring spectra using more that one technique. If one method turns out to be inappropriate, select a second method you think might be appropriate or ask your applications expert. The following methods are the most common to prepare samples : 1.) No sample preparation (example: free-standing film) 2.) Apply a thin film between 2 thin, transparent support plates (NaCl, KBr etc.). 3.) Dissolve a sample powder in a spectrophotometrically pure solvent and - apply the solution on a transparent support plate or - fill it into an IR liquid cell. 4.) Grind a powder sample into fine particles using a mortar and pestle and prepare a nujol suspension. 5.) For optically thin solid samples, grind the sample into fine particles and press the powder to obtain a very fine and even pellet. 6.) Grind a solid sample into fine particles and mix it with a transparent powder (KBr etc.). Press the powder mixture to a very thin pellet. 7.) Fill an evacuated gas cell with a gaseous sample . Gaseous samples can be injected into a gas cell (evacuated) and analyzed. The intensity of the peaks measured is influenced by the effective pathlength of the cell, the gas pressure (which is proportional to the concentration), as well as molar absorptivity. Gas cell

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Principles of ATR Since its development in the 1960s, ATR (attenuated total reflection) has been established as a standard method for both routine and research applications. ATR IR analysis can be utilized in many different fields of application without having the need of an extensive sample preparation. ATR is a reflection technique where the IR beam is directed through an internal reflection element (IRE) with a high index of refraction. The IR light is totally reflected internally off the back surface, which is in contact with the sample. The sample must have a lower index of refraction than the IRE to achieve total internal reflection. Upon reflection at the IRE/sample interface, the IR light penetrates into the sample to a small degree and the IR data from the sample is obtained.

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**(e.g. G = 38° for ZnSe for samples with n = 1.5)**

Refraction and reflection If light impinges on a boundary separating two media, the path of the light can be described by Snell’s Law. n1 • sin = n2 • sin incidence angle refraction angle N1 refraction index of medium 1 (crystal) N2 refraction index of medium 2 (sample) The bending of light occurs at the boundary where medium 1 meets medium 2. Above a certain incidence angle, the so-called critical angle (see c) in the chart), total reflection evolves where this angle G is considered to be: = 90° or: sinG = n2 / n1 (e.g. G = 38° for ZnSe for samples with n = 1.5) For incidence angles > G (see d) in the chart), light is totally internally reflected. The light beam is reflected to the original medium at the boundary between the two mediums. Upon reflection at the IRE/sample interface, the IR light penetrates into the sample to a small degree and the IR data from the sample is obtained. The electromagnetic wave that penetrates into the sample is called an Evanescent wave. ATR fiber optics utilize many internal reflections to pass light from one distance point to another. Some light is lost in passing through the crystal, where the effective length for fiber optic cables is about 1 meter in the infrared. In the NIR, fiber optic cables can be as long as 100 meters (quartz and glass are very transparent in the NIR). (Medium 1) Substance n2 (Medium 2) Crystal n1 a) b) c) d)

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** Depth of penetration Sample ATR crystal Sample n2 Refraction index**

The IR light beam penetrates the sample and the depth of penetration DP can be quantitatively described by the Harrick approximation: ATR crystal l = wavelength np = refraction index, crystal = incidence angle nsp = refraction index ratio between sample and crystal Sample ATR crystal n1 n2 Refraction index n1 > n2 Dp is defined as the distance between the sample surface and the position where the intensity of the penetrating Evanescent wave dies off to (1/e)2 or 13.5%, or its amplitude has decayed to 1/e.

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**Depth of penetration Refraction index Depth of penetration* Depth of**

Calculated depth of penetration for typical ATR crystals The depth of penetration depends on several parameters: 1.) Incidence angle: this angle is determined by the design of the ATR accessory and is constant for most ATR accessories. There are ATR accessories which have the capability to vary the angle of incidence. This can be helpful for depth profiling near the surface of a sample (within the micron range). Refraction index Depth of penetration* Depth of penetration* Material at 1,000cm-1 at 45° at 60° Diamond 2.4 1.66 1.04 Ge 4.0 0.65 0.5 Si 3.4 0.81 0.61 ZnSe 2.4 1.66 1.04 AMTIR** 2.5 1.46 0.96 *: The depth of penetration was calculated for a sample with a refraction angle of 1.4 at 1,000cm-1. **: AMTIR: Ge33As12Se55 glass

44
**Depth of penetration Refraction index Depth of penetration* Depth of**

The depth of penetration depends on different parameters: 1.) Incidence angle: this angle is determined by the design of the ATR accessory and is constant for most ATR accessories. There are ATR accessories which have the capability to vary the angle of incidence. This can be helpful for depth profiling near the surface of a sample (within the micron range). 2.) Refraction index of the ATR crystal: a higher index of refraction yields more shallow depth of penetration. ATR units with replaceable crystals can also be used for depth profiling of the sample (within the submicron range). Calculated depths of penetration for some typical ATR crystals Refraction index Depth of penetration* Depth of penetration* Material at 1,000cm-1 at 45° at 60° Diamond 2.4 1.66 1.04 Ge 4.0 0.65 0.5 Si 3.4 0.81 0.61 ZnSe 2.4 1.66 1.04 AMTIR** 2.5 1.46 0.96 *: The depth of penetration was calculated for a sample with a refraction angle of 1.4 at 1,000cm-1. **: AMTIR: Ge33As12Se55 glass

45
Depth of penetration The depth of penetration depends on different parameters: 1.) Incidence angle: this angle is determined by the design of the ATR accessory and is constant for most ATR accessories. There are ATR accessories which have the capability to vary the angle of incidence. This can be helpful for depth profiling near the surface of a sample (within the micron range). 2.) Refraction index of the ATR crystal: a higher index of refraction yields more shallow depth of penetration. ATR units with replaceable crystals can also be used for depth profiling of the sample (within the submicron range). 3.) Wavelength of light: the longer the wavelength of the incident light (lower wavenumber), the greater the depth of penetration into the sample. This yields an ATR spectrum that differs from the analogous transmission spectrum, where band intensities are higher in intensity at longer wavelength. However, the ATR spectrum is readily converted to absorbance units by selecting the “convert spectrum” option in the “manipulate” pull down menu in OPUS. 1.0 ATR Transmission 0.8 0.6 Absorbance 0.4 0.2 0.0 3,500 3,000 2,500 2,000 1,500 1,000 Wavenumbers cm-1

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**Comparing transmission and ATR measurements of a three-layer film**

The ATR technique is a surface-sensitive method due to its inherent low depth of penetration (micrometer range). An advantage to surface analysis is demonstrated by the following example: An analysis of a three-layer polymer film. The ATR spectra from the top and bottom layers of the thin film are shown to the right with the top layer at the top and the bottom layer shown on the bottom. The transmission spectrum is the middle spectrum. Quick inspection clearly shows that the top and bottom layers are different compounds and the transmission spectrum contains bands from all three layers. This ability to analyze only the surface of a sample can be a powerful tool for the chemist. Coatings on tablets, lubricants on integrated circuit boards, and thin polymer films are just a few of the many applications where ATR spectroscopy can provide information not easily obtained by any other analytical technique. Film in transmission 0.5 Film bottom side - ATR Film top side - ATR 0.4 0.3 Absorbance 0.2 0.1 4,000 3,500 3,000 2,500 2,000 1,500 1,000 500 Wavenumbers cm-1

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**Selecting an adequate ATR crystal**

Material Spectral range Refraction index Hardness*** When selecting the proper crystal for ATR analysis, sample hardness must be taken into account as well as the desired depth of penetration and spectral range. Diamond has a very high degree of hardness, but very distinctive lattice bands totally absorb between 2,500 and 1,600 cm-1. Most compounds do not have vibrations in this area. ZnSe 20, cm-1 n = 2.4 130 ZnS 50, cm-1 n = 2.3 250 Ge 5, cm-1 n = 4.0 780 Si 8, cm-1 n = 3.4 1,150 Diamond 50, ,500 cm-1 n = 2.4 9,000 1, cm-1 KRS-5* 17, cm-1 n = 2.4 40 Die häufigsten ATR-Kristallmaterialien Allgemein: IR-Transparenz, Höherer Brechungsindex als Probe Besonderer Hinweis: Härte: Diamant AMTIR** 11, cm-1 n = 2.5 170 *: KRS-5: TlI/TlBr **: AMTIR: Ge33As12Se55 glass ***: Knoop hardness

48
**Number of reflections - effective path length**

The number of reflections depends on the crystal type, the dimensions of the ATR crystal, and the incidence angle of the IR beam. A parallelogram-shaped crystal which contacts the sample on two sides can be described by: N = l / (d • tan) N = Number of reflections l = Crystal length d = Crystal thickness = Incidence angle A ZnSe crystal with a length of 80 mm, a thickness of 4 mm and an incidence angle of 45° yields N = 20 reflections. The equation for the effective path length (DE) is: DE = N • DP

49
**Selecting the adequate ATR unit**

Aside from selecting the appropriate ATR crystal type, you also have to consider the size of the crystal (number of reflections). There is a direct relationship between the number of reflections within an ATR crystal and the intensity ratio of the resulting spectrum. The higher the number of reflections, the more distinctive the bands and the better the signal-to-noise ratio. Similarly, the signal-to-noise ratio can be enhanced by increasing the measuring time. Hier nochmals die Einkanalspektren Wichtig: Diamant macht “in der Mitte dicht”

50
**Selecting the adequate ATR unit**

Multiple bounce horizontal ATR (HATR): the sample is horizontally placed onto the ATR crystal. Horizontal ATR accessories are frequently used for routine measurements, e.g. in case of high sample throughput. The ATR crystals are fixed as troughs (for liquids) or plates (for plane samples) in appropriate mountings. Simple-reflection HATR: in contrast to HATR, single reflection ATR utilizes a small ATR crystal, such that only a single reflection takes place within the crystal (good match for diamond). Single reflection ATR accessories are mainly used when analyzing small samples (> 1mm). Vertical ATR (VATR): the sample is vertically placed onto the ATR crystal. Vertical ATR is infrequently used today. Multipurpose Accessories: The ability to utilize a single accessory for ATR AND external reflection analysis can be convenient. The Bruker HELIOS and Harrick Seagull are two good examples. ATR Microscopy Modern micro ATR units are equipped with a video camera, enabling micro samples e.g. hair, fibers or samples of micro structure (printed-circuit boards) to be exactly positioned and analyzed. Samples as small as 20 microns can be quickly observed, positioned, and analyzed with the Bruker HELIOS. Hier nochmals die Einkanalspektren Wichtig: Diamant macht “in der Mitte dicht”

51
**Further applications of ATR spectroscopy**

ATR spectrometry can also be used for for sample analysis outside the sample compartment. Reaction monitoring in a demanding environment outside the spectrometer is routinely conducted utilizing ATR probe technology. 1.) ATR dipper Light is direct coupled into an ATR crystal by a flexible waveguide for analysis outside the spectrometer sample compartment. This waveguide is composed of fixed mirrors mounted in reflection coated (usually gold) light pipes. The crystal can be immersed into the reaction solution to be analyzed and monitored. 2.) MIR fiber coupling Today, one can hardly imagine NIR spectroscopy without the use of fiber optic measuring probes. Even in the middle infrared (MIR), light can also be coupled out of the spectrometer using fiber optics. However, fiber used in the MIR is not as transparent as that utilized in the near infrared and the cable length is limited to about 2 meters. Fiber coupling to 100 meters can be accomplished in the NIR. For any further information, please contact Bruker Optics:

52
**Evaluation of IR spectra**

53
Evaluation of spectra Infrared spectroscopy is an extremely efficient analytical method due to modest operating expenditure. The analytical results are provided within a short period of time without the need of extensive sample preparation. In particular, infrared spectroscopy provides data which can be evaluated by quantity as well as by quality. The following will describe the qualitative and quantitative evaluation of acquired spectra. Qualitative evaluation of spectra 1. Identify an unknown substance 2. Check the identification of a known substance Quantitative evaluation of spectra

54
**Identify an unknown substance**

a) Structural determination by interpreting spectra A functional group within a molecule is considered as a harmonic oscillator (see vibration theory) which in a first approximation vibrates without being affected by the rest of the molecule. This results in the fact that a particular functional group shows IR absorption bands within characteristic spectral ranges: this is called group vibrations. This fact serves as the basis for spectral interpretation, whereby the position, (relative) intensity and half-width of a band decide whether a band can be assigned to a specific structural group. Many functional groups of organic molecules show characteristic vibrations corresponding to absorption bands within defined ranges of the IR spectrum. These molecular vibrations are mainly restricted to the functional group and do not affect the remaining molecule, i.e. such functional groups can be identified by their absorption band. This circumstance, apart from a straightforward acquisition technique, makes IR spectroscopy to be one of the simplest, fastest and most reliable methods when assigning a substance to its specific class of compounds. The position and intensity of the absorption bands are extremely specific in the case of a pure substance. This enables the IR spectrum, similar to the human fingerprint, to be used as a highly characteristic feature for identification. 500 1,000 1,500 2,000 2,500 3,000 3,500 4,000 Wavenumber / cm-1 40 60 80 100 Transmission [%] 20

55
**IR absorption of different organic molecular classes**

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**Identify an unknown substance**

b.) Comparing with spectral libraries Besides basic spectral interpretation, various comprehensive digital spectral libraries have been compiled according to different chemical classes and groups of substance. These are provided, for example, by companies like Bruker and Sadtler. Apart from working with existing spectral libraries, it is possible to create your own libraries using modern spectroscopic software,see OPUS/SEARCH. Different spectra regarding the number of bands and half-width, may require different search algorithms. Therefore, OPUS/SEARCH has the flexibility in providing various search options.

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**identical material = identical IR spectrum**

Check the identity of a known substance Infrared spectroscopy is a perfect analytical tool for quality control. It gives the answer to the following question: “Does the quality of the raw material delivered to the receiving department comply with the specifications?” The underlying concept is very easy: identical material = identical IR spectrum The identification is done by comparing measured spectra with reference spectra already saved. The method is based upon the following considerations: chemically different materials result in different spectra real spectral differences exceed the reproducibility of repeated measurements reference samples represent the expected sample variations caused by supplier, batch, season, purity, grain size etc. It is important to note that the reference samples can vary to a certain degree, a circumstance that is experienced within quality control every day. The spectrum of the material to be identified is compared with the reference sample by means of a valid tolerance previously defined. How to create a reference library and to compare spectra will be described in the following.

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**Reference library structure**

1.) Measure reference sample 2.) Calculate average spectrum & threshold values 3.) Library structure & validation Absorbance Absorbance Wavenumber / cm-1 Wavenumber / cm-1

59
**Identifying new samples**

1.) Measure new samples 2.) Compare with library 3.) Identify material Identified sample: material X

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**Comparing spectra Euclidean distance**

Basically, to compare one spectrum with a reference library, the spectral distance between two spectra is calculated, which also defines the similarity between these two spectra. There are several algorithms available to quantify the spectral distance. Depending on the substance (or spectrum) and considering your type of problem you have to select the most suitable comparison method. The following describes the calculation of the Euclidean distance and the comparison by factor analysis. 1.) Euclidean distance The Euclidean distance is the sum of all single differences when comparing two spectra point by point, i.e. the difference between two spectra is reduced to one single numerical value. The Euclidean distance can be used very well as the measured value when comparing spectra. However, the size of this numerical value is not standardized and has always to be considered as a relative value. Generally, this method can be used for all kinds of spectra. Euclidean distance Comparing spectra “point by point”: A( i ): Absorbance value at wavelength i 1,2....i: Data points within the selected spectral range 4,000 6,000 8,000 10,000 12,000 0.2 0.4 0.6 0.8 1.0 Absorbance Wavenumber / cm-1 Acetylsalicylic acid Salicylic acid Calculating the distance: D = ((Asample(i) - Areference(i))2)1/2 D = 3.2 Difference used for Euclidean distance

61
**l2 l1 Comparing spectra Factor analysis 2) Factor analysis**

Factor analysis is a variance analysis which is widely used as a general statistical method to analyze data. It is based on the search of differences (variances) within the reference data record. Factor analysis is also called principal component analysis, PCA. The main features comprise: orthogonal data transformation substantial data compression: represents the data record by only a few latent variables Factor analysis 1. The factor loading collects the largest part of the variance in the data record 2. The factor loading collects the largest part of the remaining variance and so on … l2 l1 Advantages of factor analysis: data compression reduces considerably noise components The set of data points are represented better by a translation and rotation of the axes

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5 spectra Factor analysis The factor analysis divides spectra into factors or factor loadings and into the appropriate scores: Data matrix: n spectra with p data points Scores: d scores for each spectrum (d < n) Factors: d factors with p data points (d < n) Spectra data matrix n p = d Scores Factors 1 Factor 1 5.216 Factor 2 -0.216 Factor 3 1.73E-02 Factor 4 -1.52E-02 Factor 5 3.17E-02 2 5.95 -0.103 4.97E-04 4.33E-02 5.65E-03 3 7.731 -0.699 3.67E-04 -1.15E-02 -2.04E-02 4 5.768 0.693 2.97E-02 -3.76E-03 -1.27E-02 5 7.13 0.441 -3.75E-02 -9.54E-03 4.46E-03 Factors • Scores Factors 1 2 3 4 5 The result will be a reduced data record where the variances have been transformed into different weighted factors (factor loadings). Higher factors are generally ignored as they usually represent noise.

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Factor analysis 4,000 5,000 6,000 7,000 8,000 9,000 10,000 11,000 12,000 0.2 0.4 0.6 0.8 1.0 Absorbance Wavenumber / cm-1 Acetylsalicylic acid Salicylic acid Glucose Lactose Plot scores of factor 2 against 1 Factor 1 Factor 2 Four pharmaceutical samples are each measured by NIR spectroscopy several times. A factor analysis is performed of the spectral data. For each spectrum, the scores of the first two factors are plotted. This example shows how a complex set of data can be reduced to a relatively simple 2D plot.

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Scores plot Comparing spectra “point by point” by means of scores between spectra or between spectra and average spectrum The MAHALANOBIS distance between spectra is corrected for collinearity so different directions in a scatter plot are weighted according to the variance (below: 2D plot of T1 against T2, with contours of equal Mahalanobis distance to the center) For the calculation of the spectral distance D between two spectra, the score coefficients T are used from the factor analysis: T2 T1 Ti Score of factor i 1,2....i Factors used

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**Quantitative evaluation of spectra**

The basic principle for quantitative evaluation in optical spectroscopy as well as in IR spectroscopy is the Bouguer-Lambert-Beer Law which had already been defined in Quantitative determinations by means of IR spectroscopy are preferably performed in solution. Transmission T of a sample is defined as: T = I / I0 Io is the intensity of the incident light beam, I is the intensity of the light beam leaving the sample. The percentage transmission (%T) is 100 • T. When traversing the measurement cell, the light intensity decreases exponentially: I = I0 • exp(-2.303 • c • b) Where is the molar absorption coefficient (in L mol-1 cm-1), c is the sample concentration (in mol L-1) and b the thickness of the measurement cell (in cm). The absorption coefficient is a value which depends on either the wavelength or the wavenumber, which is typical for the compound analyzed. From the equation above, it follows that: log (I / I0) = - • c • b, or: A = log (I0 / I) = • c • b where A is the absorbance. Because of the Bouguer-Lambert-Beer Law, the relationship between absorbance and concentration of the absorbing substance is a linear function.

66
**Quantitative evaluation of spectra**

In practice the relationship between concentration and absorbance is empirically determined by calibration. Calibrating means finding the mathematical connection between concentration and the measurement values. In the first step, spectra of substances with known composition are recorded. Then, these acquired spectra and the data available from a reference analysis (concentration or substance properties) are used to determine a calibration function. The software package OPUS/QUANT provides several algorithms to do this. In the second step, spectra of substances with an unknown composition are measured and then used to determine the properties of interest by means of the calibration function. There are two different forms of calibration: Univariate calibration (OPUS) Correlates just one piece of spectral information (e.g. peak height or peak area) with the reference values of the calibration set. Multivariate calibration (OPUS/QUANT) Correlates considerably more spectral information using larger spectral ranges with the reference values of the calibration set. This leads to a higher degree of precision and reduced chance of error. Partial Least Squares ((PLS) is an example of this method and is implemented in OPUS/QUANT. 1 3 2 4 Absorbance Wavelength Calibration X Analysis 1 2 3 4 Absorbance Concentration X

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**PLS regression M) 2) 1) W1 W2 WN X 1 2 M W C1 C2 CL C L Y N . = . . .**

In PLS, the calibration involves correlating the data in the spectral matrix X with the data in the concentration (or properties) matrix Y. The X and Y matrices are reduced to only a few factors using all of the available information. The final model consists of a score matrix for X and a score vector for Y (in PLS-1) which are linearly related. This means that the factoring of the spectral data is more suited for concentration prediction. M) 2) 1) . W1 W2 WN Wavelength Signal X = 1 2 M W N C1 C2 CL C L Y

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**Practical considerations**

1.) Choice of concentration range The concentration range of the calibration set should be larger than the range of the samples to be analyzed. Analysis value 2.) Choice of component values Typical concentration range Individual component values should uniformly cover the complete concentration range In the case of multi-component mixtures, the individual concentration values must not decrease or increase to the same degree (avoid collinearity) Reference value

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**Practical considerations**

3.) Number of calibration samples Simple systems: ~ at least 20 samples Typical systems: ~ samples Complex systems: ~ 150 samples 4.) Selecting adequate spectral ranges for calibration Avoid spectral noise at the beginning and at the end of the spectrum, where for example there may be a detector or window cutoff Avoid ranges of total absorbance ( > 2.5 AU) Use large spectral ranges, if possible for calibration (“Full Spectrum Calibration”) 5.) Parameters influenced by quality Quality of the measuring instrument (resolution capability, stability, S/N-ratio, measuring precision, robustness, ...) Quality of reference data (frequently the quality of an IR/NIR method is almost exclusively determined by the quality of the reference data).

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If you have any further questions about IR spectroscopy, please contact the application team of Bruker Optics: North America: Bruker Optics Inc 19 Fortune Drive Billerica, MA 01821, USA Phone: Fax: Europe: Bruker Optik GmbH Rudolf-Plank-Str. 27 76275 Ettlingen, Germany Phone: Fax: Asia: Bruker Optik Asia Pacific Ltd. Suite 1116 Nan Fung Commercial Centre 19 Lam Lok Street, Kowloon Bay, Hong Kong Phone: Fax:

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Optical sources Lecture 5.

Optical sources Lecture 5.

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