Modern Fourier Transform Infrared Spectroscopy (Comprehensive Analytical Chemistry)

Series Editor's Preface This book on Modern Fourier Transform Infrared Spectroscopy is a useful addition to the ComprehensiveAnalytical Chemistry series. The work contains different chapters that cover both fundamental and applied aspects of infrared spectroscopy. Particular attention is given to fundamentals of vibrational spectroscopy and to the recent developments of hyphenated chromatographic techniques. In addition, the major portion of the applications described in this book deal with polymeric and biological materials. Chemometric interpretation and data analysis are also described in detail in the last chapter of the book, indicating their relevance in infrared spectroscopy. The book can be used as an academic text and as reference book either for those with more expertise or for those starting with this technique. Overall, the book covers an important technique increasingly used in analytical chemistry. Finally I would like to thank the authors of the book for their time and efforts in preparing their contributions. Without their engagement this reference work on infrared spectroscopy would certainly not have been possible. D. Barcel6

xvii

Acknowledgements

Writing a book needs a lot of reading, careful planning and writing. The process is time consuming and requires the assistance of people on whom we can rely. During the process of writing this book many have helped us with physical work, ideas and support. It is not possible to thank everyone who has contributed to the book. However, there are some who have contributed to elevate the quality of the book and we are grateful for their efforts. In this respect, we would like to thank Professor Rolf Manne, Department of Chemistry, Agder University College for critically reviewing Chapter 4. We would like to thank Dr. Hideki Kandori (Kyoto University) for critically reviewing parts of Chapter 8 and Ms. Seiko Hino for polishing the English in Chapters 3, 5, 7, and parts of Chapters 8 and 9. In addition one of the authors (VGG) would like to thank Sheila E. Rodman of Polaroid Corporation for the collection of some of the materials used in Chapters 6, 7 and 8. Finally, we would like to thank our families and friends who have given us moral support and helped us through some difficult times during the writing of the book.

xviii

Authors' Preface Infrared spectroscopy has a history of more than a century: the characteristic absorptions of functional groups in the infrared region were known even in the 19th century; the first infrared atlas was published in 1905, twenty years before the birth of quantum mechanics. However, even though infrared spectroscopy is a relatively old technique, it has always been a popular technique for chemical analysis. Developments in computer technology, sensitive detectors and accessories for new sampling methodologies in the infrared region have made infrared spectroscopy one of the most powerful and widespread spectroscopic tools of the 20th and 21st centuries. The applications of infrared spectroscopy, and of Fourier transform infrared spectroscopy (FT-IR) in particular, are ever expanding, due to its versatile nature. The enormous number of articles and research papers published every year that deal with infrared spectroscopy and its applications is clear evidence to this. The book "Modern Fourier Transform Infrared Spectroscopy" has been written to reflect the popularity of infrared spectroscopy in several different fields of science. The chapters are designed to give the reader not only the understanding of the basics of infrared spectroscopy but also ideas on how to apply the technique in these different fields. The book is suitable for students at graduate level as well as experienced researchers in academia and industry. The first three chapters deal with the fundamentals of vibrational spectroscopy. Since spectroscopy is the study of the interaction of electromagnetic radiation with matter, the first two chapters deal with the characteristics, properties and absorption of electromagnetic radiation. Chapter 3 provides the basis for vibrations in molecules from both classical and quantum mechanical points of view. The absorption of infrared radiation by a vibration in a molecule depends on the symmetry of the molecule and the symmetry of the vibrations. As this aspect is not usually treated in textbooks on infrared spectroscopy, Chapter 4 deals with the symmetry aspects of molecules and illustrates how the reader can determine the vibrations that are infrared active. Chapter 6 is an overview of the instrumentation used to perform the majority of Fourier transformed infrared spectroscopic experiments xix

today. The chapter starts with an overview of the history of FT-IR spectroscopy from the construction of the first interferometers in 1880 to the present day and continues with a description of the components of an interferometer and the various scanning techniques (continuousscan and step-scan). Chapter 7 first describes sampling techniques used in transmission and reflection spectroscopy and then a variety of the so-called hyphenated techniques that combine the use of FT-IR spectroscopy with another analytical technique. Thermogravimetric analysis (TGA/FT-IR), liquid chromatography (LC/FT-IR), gas chromatography (GC/FT-IR) and supercritical fluid chromatography (SFC/FTIR) are the combinations discussed in this book. Chapter 8 depicts certain applications of FT-IR spectroscopic techniques to basic and industrial research. Specifically, a large portion of the chapter deals with the characterization of polymers and polymeric surfaces, whereas the remaining part describes applications to organic thin films and biological molecules. One subcategory treated in detail is the determination of molecular orientation in polymers via static and dynamic FT-IR experiments. Another subcategory is the applications that involve optically active materials and conducting polymers. In addition, very significant developments have recently taken place in the area of infrared microspectroscopy and especially in infrared imaging with the introduction of focal plane array detection. Part of this chapter is dedicated to an explanation of the experimental procedures associated with these imaging experiments along with selected examples from the recent literature. Finally, Chapter 9 deals with some modern analytical methods in infrared spectroscopy. Again, the methods described here are not very common in books on infrared spectroscopy. The first part of the chapter deals with chemometric techniques that can be applied to semi-quantitative and quantitative analysis of infrared spectroscopic profiles. The text is designed to give the theoretical basis of these techniques and in particular how they can be applied to infrared spectroscopic data profiles. In this chapter, the subject of two-dimensional correlation spectroscopy (2D-IR) is also discussed. The principles of the technique along with selected examples of the applications of the 2D-IR treatment are presented. Alfred A. Christy Yukihiro Ozaki Vasilis G. Gregoriou April 2001 xx

Chapter 1

Electromagnetic radiation and the electromagnetic spectrum

We have come across people talking about microwave, UV radiation, radio waves, x-rays, radar, cosmic rays and so on. We understand the dangers of UV radiations from the sun and the use of microwave in heating food. What do we associate with all these different terms? We understand without learning the physics of these different radiations that they are associated with different energies. For example, white light is a form of energy and it comprises a mixture or spectrum of seven different colours, which are all visible to the human eye. All these colours of which white light is composed have different energies in the descending order: violet, indigo, blue, green, yellow, orange, red. A photographic plate is readily affected by violet light, unlike red light which has almost no effect. From the above discussion, it is clear that the spectrum of different radiations falls into a larger scale of spectrum where white light is a very small part. The larger scale containing the spectrum of different radiations is called electromagneticspectrum. The physical properties of radiations cannot be explained by a single theory. Some properties such as propagation of radiation through a medium, diffraction and reflection are better explained by a theory called wave theory and properties like momentum are better explained by particle theory. The propagation of radiation through space involves electric and magnetic components of the radiation and hence the term "electromagnetic". Spectroscopy generally deals with the interaction of electromagnetic radiation and matter. In order to understand this interaction, we must understand the characteristics of the electromagnetic radiation and the matter involved in the interaction. 1

1.1 WAVE NATURE OF ELECTROMAGNETIC RADIATION-WAVE CHARACTERISTICS AND WAVE PARAMETERS According to electromagnetic theory, electromagnetic radiation is a form of energy that is composed of oscillating electric and magnetic fields acting in planes that are perpendicular to each other and to the direction of propagation (Fig. 1.1). The oscillating electric field is simple harmonic and propagates as waves with a velocity (c) 3x10 8 m s- in vacuum. The propagation velocity varies with the refractive index of the medium. In a two-dimensional representation, the variation of the electric field strength of an electromagnetic radiation with propagation time in space can be paralleled to the variation of the y co-ordinate of the trace of a particle moving around a circle of radius A with a constant angular velocity o radians per second (Fig. 1.2). Let us assume that OX and OY represent the positive directions of the x and y axes and O represents the origin of these axes. Let us also assume that at time zero the particle passes through x and then consider the position of the particle after t seconds. The angle traversed by the particle in t seconds is ot radians. The y co-ordinate of the position of the particle can be given by equation (1.1)

y =A sin ot

Electric field streng~

Direction of propagation

Fig. 1.1. Oscillating electric and magnetic fields of electromagnetic radiation.

2

ct=r/2

s-1

act=r

wt= 3r12 Fig. 1.2. Trace of a particle moving around a circle of radius A with an angular velocity rads- 1.

At time zero (i.e. ot = 0) the particle is at x and this function is minimum with a value 0. At time 7d/20 (i.e ot = c/2) the particle is at y and the function is maximum with a value A, the amplitude of the function. At time nl/c (i.e cot = g), the particle is at Z and the function has another minimum. Similarly the function will have another maximum and another minimum at times 3/2co (cot = 37/2) and 2/co (ot = 2), respectively. The particle takes 2/o seconds (remember one circle is 2i7 radians) to complete the journey through the circular path once (one cycle). This is called the period of the motion and denoted by . The number of cycles the particle traces through in one second is o/2t {1/ (2t/o)} and is called the frequency (v) of the motion. The frequency is abbreviated by the symbol Hz. The angular velocity of the motion can then be written in terms of the frequency as

co = 2v

(1.2)

A two-dimensional plot of the variation of the y co-ordinate of the position of the particle with time is shown in Fig 1.3. Equation (1.1) can be written to include the frequency of the motion and time as y =A sin 2vt

(1.3) 3

A

O.5 A

>1

A

0--

-0.5A

-A

Time in seconds

Fig. 1.3. A two-dimensional plot of the variation of the y co-ordinate moving in a circle as shown in Fig. 1.2.

0.SA 0

-0.5A

-A

Propagation distance in metres

Fig. 1.4. The propagation of electromagnetic radiation.

The propagation of the electromagnetic radiation in space is 3x108 m s l. Figure 1.3 can be redrawn (Fig. 1.4) in a similar manner to represent the distance of propagation of electromagnetic radiation along the x axis. When the x axis represents the distance, we can define some other parameters characteristic to electromagnetic radiation. In the 4

preceding discussion, we learnt that the y co-ordinate of the particle has zero values at times 0, n/co and 27/co. That is at distances 0, rc/lo and 2nc/o, the propagating electric field of the electromagnetic radiation will have zero field strength. But the distance between the extreme points, that is, the distance travelled during a complete cycle of oscillation is called wavelength (k). However, because of the symmetry of the sine wave, the wavelength can be defined as the distance between two similar points in the wave. The wave has a frequency v and therefore the velocity of propagation can be written as kv = c (in metres per second)

(1.4)

has dimension m (metre). The This implies that the wavelength inverse of the wavelength, when the wavelength is expressed in centimetres is called wavenumber and is denoted by v. The dimension for wavenumber is cm -1 . v = 1/k cm -1

(1.5)

The wavenumber can also be thought of as the number of waves in 1 cm length. The above two equations combined give Eq. (1.6). v = Vc (c in centimetres per second)

(1.6)

The propagation distance s of an electromagnetic radiation in t seconds is given by s = kvt = ct

(1.7)

Equation (1.7) can be combined with Eq. (1.1) to give a relationship describing the variation of the field strength of an electromagnetic radiation with the distance of propagation and its wavelength as follows y =A sin (2nrs/k)

(1.8)

The flux of energy of an electromagnetic radiation along the direction of propagation is equally divided between electric and magnetic fields. Electromagnetic radiations are produced by accelerating electric charges and electric charge accelerations are produced when 5

electromagnetic radiations are absorbed. Electromagnetic radiations of different energies are produced when the energy involved in the acceleration of electric charges is different. We shall see later that y-rays to microwaves are produced during oscillations of charges at different states in matter.

1.2 QUANTUM CONCEPT AND PARTICLE NATURE OF ELECTROMAGNETIC RADIATION Not all properties of electromagnetic radiation can be explained by the wave theory. Electromagnetic radiation has particle-like (corpuscular) properties in addition to the wave properties. The corpuscular nature of electromagnetic radiation was developed during the early 20th century in order explain certain characteristics that classical physics failed to explain. Max Planck originated the quantum hypothesis to explain the discontinuity in the energy of an oscillator of frequency v. Planck's hypothesis suggests that the energy of an oscillator with frequency v is not continuously variable, but restricted to integral multiples of hv as 0, lhv, 2hv, 3hv, ... nhv where hv is a quantum of energy and h is a universal constant which is known as Planck's constant. Albert Einstein, in 1905, related Planck's quantum hypothesis to the photoelectric effect where electrons from metal surfaces were ejected by ultraviolet radiation. Einstein explained that the electrons from a metal are ejected when they receive energy at least equivalent to their binding energy. Furthermore, the energy of the radiation must be confined to a small region of space in order to transfer the energy entirely to an electron and eject it instantaneously. The ejection will not be instantaneous if the energy is spread evenly across the entire wave front. This leads to an understanding that the energy of electromagnetic radiation can be considered as packets (quanta) of energy hv (Fig. 1.5). The quantum of radiant energy was named photon by Lewis in 1926. The corpuscular nature of photons was explained by their possession of momentum. The relativistic expression shown in Eq. (1.9)

Fig. 1.5. Energy packets (quanta) of photons. 6

leads to a value of hv/c for linear momentum (p) of a photon which has no mass (m = 0). E = m2c 4 +p2c 2

(1.9)

The scattering of photons (electromagnetic radiation) during collisions with electrons is a clear indication that the photons possess momentum.

1.3 UNITS OF WAVE PARAMETERS The SI unitary system requires that the dimensions of length, mass and time are specified in metres (m), kilograms (kg) and seconds (s), respectively. Table 1.1 shows the units of some parameters and constants we have come across during our discussion. TABLE 1.1 Symbols and dimensions of wave parameters and related terms Parameter Symbol Dimensions of units Energy

J (Joule)

kg m2 s- 2

Wavelength Frequency Velocity of light

Hz (Hertz) 3x108

m s-1 (cycles per second) m s- 1

Planck's constant

6.626x1034

Js

1.4 ORIGIN OF ELECTROMAGNETIC RADIATION AND ELECTROMAGNETIC SPECTRUM: y-RAYS TO MICROWAVE The origin of electromagnetic radiation varies widely. The universe contains radiations ranging from y-rays to microwaves. A system that emits radiation is also capable of absorbing that radiation. An example of this is the absorption and emission of yellow light by sodium atoms by which sodium is quantitatively determined by atomic spectrometry. We use a sodium lamp that emits light at a particular frequency which 7

visible infrared radiation Y-rays Fig. 1.6. Oscillators in an atom and a molecule.

is produced when an excited electron relaxes from an excited state to the ground state to excite an electron from a ground state to the same corresponding excited state in another sodium atom. As mentioned earlier, the electromagnetic radiations originate from oscillators arising from the acceleration of electric charges. Gammarays are produced by the oscillators in the atomic nuclei; x-rays are produced by the oscillators arising from the tightly bound electrons in the vicinity of the nucleus of atoms; visible and ultraviolet radiations are produced by the oscillators arising from the outer electrons in the molecules and atoms; infrared radiation is produced by the oscillators arising from the vibration and rotation of molecules. Now we should be able to understand why electromagnetic radiations of different dimensions are used in studying nuclear, electronic, vibrational and rotational transitions in matter. TABLE 1.2 Electromagnetic spectrum: yrays to radio waves, their approximate frequency and energy range Radiation Wavelength (m) y-rays x-rays

5x10-12_10

-1 2

10-S-5x10

- 12

Frequency range (Hz)

Energy range per photon (J)

Energy range per mole (kJ)

6xl103x1O20

-1 4x10-12x10

24x106-1.2x10 s

3x10

8

16

-6x101 9 5

16

3

18

2x 10- -4x 10 1 8

- 14

1200-24x10 6

Far UV

1.8x10 7-10

Near UV

3.5x10- 7-1.8x10-

Visible

7.7x10-7-3.5x10- 7 3.9x1014-8.6x1014 2.58x10-19-5.7x10 l- 9 155-343

Near IR

4 4 0 19 2.5x10-6-7.7x10 - 7 1.2x10' -3.9x101 8x10-2 2.58x10

Mid IR

5x10-5-2.5x106

6x 1012 -1.2x1014

10-3-5x105

1 3x10 -610

Far IR

Microwave 3x10-1 -10

8

-3

1.7x 101 -3X10 7

1.13x10

-2x10 18

680-1200

8.6x1014-1.7x1015 5.7x10-19-1.13x 10-18 343-680

10 9-3x10 11

2

4x10-2_8x 10 22

2x10- -4x10

-2 1

6.6x10-2-2x10 - 2 2

48-155 2.4-48 0.12-2.4 0.0004-0.12

Electromagnetic radiation spectrum in the larger scale is given in Table 1.2. It is difficult to define precise limits between different divisions and the reader should understand that the divisions are approximate and may slightly vary with the limits given in other textbooks.

GENERAL BIBLIOGRAPHY P.W. Atkins, Molecular Quantum Mechanics. Oxford University Press, London, 1984. C.N. Banwell and E.M. McCash, Fundamentals of Molecular Spectroscopy. McGraw-Hill, London, 1994. W. Kemp, OrganicSpectroscopy. Macmillan, London and Basingstoke, 1978. J.H. Vander Maas, Basic InfraredSpectroscopy. Heydon & Sons, London, 1972.

9

Chapter2

Interaction of electromagnetic radiation with matter

2.1 ABSORPTION OF ELECTROMAGNETIC RADIATION Matter is composed of molecules or atoms. Atoms are composed of nuclei containing protons, neutrons and electrons surrounding the nucleus. This means that matter is full of oscillators of very different dimensions. Any of these can be excited to a higher level using electromagnetic radiation of appropriate energy. For example, vibration in a molecule containing two atoms is equivalent to a simple oscillator. This vibration can be excited to the next vibrational level by irradiating the molecule with infrared radiation of appropriate energy (i.e. appropriate frequency). Here, we shall just assume that absorption takes place (Fig. 2.1) without considering the conditions for infrared absorption, which we shall consider in Chapter 3. If the energy of the radiation does not match the energy difference between the excited and ground states of the molecule, no absorption will take place. If the frequency of the radiation that is absorbed by the molecule is v, then the energy difference between the ground state and excited state is given by the Eq. (2.1). Ee-Eg = AE = hv =hc =hvc

(2.1)

where Ee, Eg are absolute energies of the excited and ground states. For example, if the wavenumber of the radiation needed to excite a vibrational mode in a molecule is 3000 cm- l, then the energy absorbed by one molecule is

11

I

-

E x.tedx

AE = hp

Fig. 2.1. Absorption of electromagnetic radiation.

hvc = 6.626x 10 -3 4 J s x3000 cm - l 3x 101 cm s- l = 5.963x10 - 2 0 J

AEmoiecule =

(2.2)

If we are interested in the energy absorbed by a mole of the substance then we have to multiply the answer by the Avogadro's number NA = 6.023x10 2 3 mol-1. That is, the energy absorbed by a mole of the substance is AEmole

5.963x1020 J x 6.023x1023 mol - = 35.9x10 J mol-' = 35.9 kJ mol=

(2.3)

2.2 PRESENTATION OF DATA: A LINE SPECTRUM

In absorption spectroscopy, the information sought for is the absorption of radiant power from a source by the sample. In infrared spectroscopy, this is done in practice by a spectrometer; the construction of this instrument will be dealt with in a later chapter. The idea behind the technique is to send radiation through (or interact with) a sample and measure the characteristics of the radiation emerging from the sample. Let us assume that monochromatic radiation (only one-frequency radiation) with sufficient intensity and of a frequency that matches the frequency of absorption is passed through a sample. A part of the radiation will be absorbed and the intensity of the radiation emerging from the sample, the transmitted radiation, will be of reduced 12

100

0

v

Frequency, Hz Fig. 2.2. A transmittance spectrum.

1

aW

0

V

Frequency, Hz

Fig. 2.3. A line spectrum of absorption.

intensity. radiation expected Similarly

If we plot the percent of the intensity of the transmitted relative to the source intensity against frequency, then the plot should contain a single line as shown in Fig. 2.2. an absorption spectrum will be as shown in Fig. 2.3.

2.3 LINE BROADENING IN INFRARED ABSORPTION SPECTROMETRY The real absorption spectrum of a diatomic molecule, which has only a single vibrational mode, is not a line spectrum, but an absorption spectrum containing a broad peak with a maximum at the frequency of absorption as shown in Fig. 2.4. The line broadening in spectrometry arises because of characteristics adherent to molecules in different phases and uncertainty in the 13

1

Q -E 11

ii A

V

Frequency, Hz

Fig. 2.4. A broad peak representing an absorption.

energy levels due to the limited residence time of the particle in the excited state. In solids, liquids and gases the particle velocities are different. The particles collide more frequently in the gas phase than in the liquid phase. The vibrational and rotational energy levels are perturbed from their actual values and lead to small variations in the ground state energy levels. This implies that solids should give sharp bands. This is true, but the bands are split because of electronic interactions. Spectral line broadening arises also due to the Doppler effect. The infrared measurements are made in cells where the radiation is allowed to pass through or interact with the sample. Molecules that are in motion towards the source will absorb radiation of higher frequency compared with the molecules that are moving away from the source (lower frequency). One of the important effects that cause spectral line broadening is the uncertainty effect. The uncertainty principle proposed by Werner Heisenberg suggests that there is natural limitation on how precise a pair of physical parameters can be made. In vibrational spectrometry the uncertainty in the energy level of the excited state, AE, and uncertainty in the lifetime of the molecule at an excited vibrational state, At is related by the uncertainty relationship as follows. AE At 2 h/2

(2.4)

This can be rewritten to include frequency v as follows A(hv)At 2 h/2t 14

(2.5)

i.e. (2.6)

AvAt > 1/2xT

If the residence times at the excited vibrational states are infinite then the uncertainty in the frequency of absorption is zero and the frequency can be determined precisely. The molecules that undergo vibrational transitions have a finite time of residence of 10-8 s and this leads to an uncertainty in the frequency Av > h/(2xAt) = 1/(2x3.14x 10-8) s

108

(2.7)

This uncertainty is small compared to the radiation frequency of infrared radiation 1012-104. This leads us to conclude that absorption by a single vibrational mode will be a band spectrum with a finite frequency width as shown in Fig. 2.4.

2.4 MEASURED SPECTRA OF DIATOMIC MOLECULES The measured infrared spectra of diatomic molecules do not show a single band as we expected but fine spectra with several fine peaks spaced at equal intervals with a space in the middle. The fine spectra arise from the rotational spectra of diatomic molecules. When a diatomic molecule is excited the rotational levels are also excited and the rotational absorptions are superimposed on the vibrational spectra of the diatomic molecules. The vibrational spectrum of carbon monoxide at high resolution is shown in Fig. 2.5.

2.5 NORMAL OR FUNDAMENTAL VIBRATIONS Molecules of a substance are in continuous motion. They move (translation) and rotate. Each atom in a molecule assumes a new position with time. Each atom in a molecule can be represented by three co-ordinates (x, y, z) in a Cartesian system. We say that the atom has three degrees of freedom. When we consider a molecule containing N atoms, the atoms have a total of 3N degrees of freedom. The result of the movements of 15

o o

-

C o

g

C g

O

C o

C 0

o -~ CC O :1

C, 0O. o

O

UC

C o

-E:

oC CC

N

g

N

E

o

-~C rl

i CC M

r

c I

F,

the individual atoms can be represented by the movement of the centre of mass of the molecule. The position of the centre of mass-the position of the molecule in space-can then be represented by three co-ordinates, i.e., three degrees of freedom. The molecule rotates about its centre of mass, and the rotations about three mutually perpendicular axes passing through the centre of mass require three more degrees of freedom. The free movement of the atoms is restricted by the bonds between the atoms, but they vibrate from their equilibrium positions. These are represented by the remaining 3N - 6 degrees of freedom. These vibrational motions are called normal or fundamental vibrations. Linear molecules need only two degrees of freedom to specify rotations of the molecules in space. Therefore, the linear molecules have 3N - 5 fundamental vibrational modes.

2.6 INFRARED SPECTRUM OF POLYATOMIC MOLECULES Fortunately, the energy needed to excite most of the vibrational modes in organic molecules falls in the infrared spectral frequency region 7.5x10 1 2-1.2x101 4 (250-4000 cm-l). In infrared spectroscopy, it is tK 1350 f- Fierprt--

- GroYp requencies -

3.0-

900

2.5 i 2.0

eI n.

w 1.5 1.0.

0.5. 0.05 4000.0

iI

Y

I1}

AAA,

.

.

3000

2000

)L

i

I1

vI

,

.

1500

, Aid VCLU v

·.

X

I

kA

- V

I

.

1000

,

..

450,0

Wavenumber cm-1

Fig. 2.6. An infrared absorption spectrum of polyatomic molecule (polystyrene).

17

customary to give the excitation energy needed in terms of wavenumber v which has a directly proportional relationship with frequency v as v = v/c. The region 4000-1350 cm-l contains group frequencies and the region 1350-900 cm-l contains low energy vibrations. This region is characteristic of the molecule and is called the fingerprintregion. If the molecule is polyatomic and the radiation is polychromatic, then the vibrational modes absorbing in the region 4000-250 cm-l will result in considerable overlap and the plot between absorbance and wavenumber will be a mixture of sharp and broad bands over the whole range. Wavenumbers are usually plotted from higher wavenumbers to lower wavenumbers and the plot is called an infrared absorption spectrum (Fig. 2.6).

GENERAL BIBLIOGRAPHY P.W. Atkins, Molecular Quantum Mechanics. Oxford University Press, London, 1984. C.N. Banwell and E.M. McCash, Fundamentals of Molecular Spectroscopy. McGraw-Hill, London, 1994. W. Kemp, Organic Spectroscopy. Macmillan, London and Basingstoke, 1978. J.H. Vander Maas, Basic InfraredSpectroscopy. Heydon & Sons, London, 1972.

18

Chapter 3

Theory of infrared spectroscopy

3.1 PRINCIPLES OF INFRARED SPECTROSCOPY When irradiated with infrared light, a molecule absorbs it under some conditions. The energy hv of the absorbed infrared light is equal to an energy difference between a certain energy level of vibration of the molecule (having an energy Em) and another energy level of vibration of the molecule (having an energy En). In the form of an equation, hv=En-Em

(3.1)

holds. In other words, absorption of infrared light occurs principally based on a transition between energy levels of molecular vibration. This is why an infrared absorption spectrum is a vibrational spectrum of a molecule. Satisfying Eq. (3.1) does not always cause infrared absorption. There are transitions which are allowed by a selection rule (i.e., allowed transition) and those which are not allowed by the same rule (i.e., forbidden transition). In general, transitions with a change in the vibrational quantum number by +1 are allowed transitions and other transitions are forbidden transitions. This is what is known as a selection rule with respect to infrared absorption. Another selection rule with respect to infraredabsorption is one which is defined by the symmetry of a molecule. This selection rule is, in other words, "a rule that infrared light is absorbed when the electric dipole moment of a molecule changes as a whole in accordancewith a molecularvibration." The two selection rules are developed from quantum-mechanical considerations. According to quantum mechanics, for a molecule to transit from a certain state m to another state n by absorbing or 19

emitting infrared light, it is necessary that the following definite integral: (P)mn = f W.tln

mdQ

(3.2)

or at least one of (py)mn, and (z),, which are expressed by a similar equation is not 0, where Px denotes an x-component of the electric dipole moment; v denotes the eigenfunction of the molecule in its vibrational state; and Q denotes a normal coordinate (i.e., a normal vibration; see Section 3.3) expressed as a single coordinate. Now, let us consider only (px)mn. A distribution of electrons in the ground state changes as the coordinate expressing a vibration changes and, therefore, the electric dipole moment is a function of the normal coordinate Q. Hence, Px can be expanded as follows: PX =(

)o (d

/Q)oQ+

1(82 2

/Q2)Q2+...

(33)

Expressed by a displacement of atoms during the vibration, Q has a small value. This allows us to omit Q2 and the subsequent terms in the equation above. Substituting the terms up to Q ofEq. (3.3) in Eq. (3.2) 8 (p.x)m, =(Pi)oWVm dQ+

x

f

QydQ

(3.4)

is obtained. Due to the orthogonality of the eigenfunction, the first term of this equation is 0 except when m = n holds. The first term denotes the magnitude of the permanent dipole of the molecule. For the second term to have a value other than 0, both (p /taQ)o# 0 and lJynQ /mdQ • 0 must be satisfied. These two conditions lead to the two selection rules. The nature of the eigenfunction permits the integral to have a value other than 0 only when n = m 1 holds. (Considering Q2 and the subsequent terms of Eq. (3.3) as well, we can prove that even when n = m + 1 fails to hold, (x)mn has a value, even though small, other than 0). The first selection rule regarding infrared absorption is thus proved. The other selection rule, which is based upon the symmetry of a molecule, is obtained from (JcdQ) 0 • 0. The relationship (pJaQ)o X 0 indicates that infrared absorption occurs only when certain vibration changes the electric dipole moment. The vibration is infrared active 20

when (pJaQ) 0

0 holds, but is infrared inactive when (p 1 JcaQ)0 = 0

holds. Since most molecules are in the ground vibrational state at room temperature, a transition from the state v" = 0 to the state v" = 1 (first excited state) is possible. Absorption corresponding to this transition is called the fundamental. Although most bands which are observed in infrared absorption spectra arise from the fundamental, in some cases, we can find bands which correspond to transitions from the state v" = 0 to the state v" = 2, 3, 4 ... (i.e., overtone transitions). However, since overtones are forbidden, overtone bands are very weak. The horizontal axis and the vertical axis of an infrared absorption spectrum must now be explained. A frequency is indicated along the horizontal axis in the units of wavenumber (cm-1 ) (with higher wavenumbers always on the left-hand side) (Fig. 3.1). On the other hand, a transmittance T (%) (Fig. 3.1a) or an absorbance E (Fig. 3.1b) is expressed along the vertical axis. While infrared spectra include reflectance spectra and emission spectra in addition to absorption (a)

80 60 40 20 0 1.0 (b) 0.8 0.6 0.4 0.2 0

4000

3200

2400

1600

800

Fig. 3.1. Examples of infrared spectra: (a) transmittance spectrum; (b) absorbance spectrum. 21

spectra, a reflectance, an emission intensity, etc. are expressed along the vertical axis in the case of a reflectance spectrum, an emission spectrum, etc.

3.2 CHARACTERISTICS OF INFRARED SPECTROSCOPY Infrared spectroscopy provides detailed information about vibrations of a molecule. Since molecular vibrations readily reflect chemical features of a molecule, such as an arrangement of nuclei and chemical bonds within the molecule, infrared spectroscopy contributes considerably not only to identification of the molecule but also to study of the molecule structure. Furthermore, an interaction with a surrounding environment also causes a change in molecule vibrations, and hence, infrared spectroscopy is useful in studying the interaction, too. Infrared spectroscopy has many uses from basic research to various applications. Why is infrared spectroscopy useful? The answer is simple. It is spectroscopy which probes a vibration of a functional group. Infrared spectroscopy can be used not only for the identification of a functional group, but also for the investigation of the chemical bond and environment of the functional group. For example, C=O groups of -C=C-C=O and -CH2 -CH 2 -C=O give different frequencies. Of course, a -C=O group and -C=O .. H-O- also yield different frequencies. We can summarize the characteristics of infrared spectroscopy as follows: 1. Using an electromagnetic wave having a low energy, infrared spectroscopy rarely damages a sample. In addition, infrared spectroscopy may be used for non-destructive analysis of a sample. 2. Infrared spectroscopy is applicable to a sample in various states, e.g., solid, crystal, fibre, film, liquid, solution and gas. Furthermore, measurements of infrared spectra of a sample in a solution and in a solid state, allow us to compare its structure in the solution with that in the solid. 3. Infrared spectroscopy uses not only so-called infrared absorption, but can utilize infrared reflection, emission, photoacoustic spectroscopy as well. 4. Connection with an optical microscope, a gas chromatograph, a liquid chromatograph or other instrument is relatively easy, which allows hyphenated analysis. 22

3.3. MOLECULAR VIBRATIONS Knowledge about vibrations of a molecule is crucially important for understanding infrared spectroscopy. It is useful also for Raman and near-infrared spectroscopy. While vibrations of a polyatomic molecule are generally complex, according to harmonic oscillatorapproximation (i.e., an approximation on the premise that the force which restores a displacement of a nucleus from its equilibrium position complies with Hooke's law; vibrations in harmonic oscillator approximation are called harmonic vibrations), any vibrations of the molecule are expressed as compositions of simple vibrations called normal vibrations. Normal vibrations are vibrations of nuclei within a molecule, and translational motions and rotational motions of the molecule as a whole are not included in normal vibrations. In each normal vibration, all atoms vibrate with the same frequency (normal frequency), and they pass through their equilibrium positions simultaneously. In general, a molecule which consists of N atoms has 3N - 6 normal vibrations (3N - 5 normal vibrations if the molecule is a linear molecule). Since normal vibrations are determined by the molecular structure, the atomic weight and the force constant, when these three are known, we can calculate the normal frequencies and the normal modes.

3.4 A VIBRATION OF A DIATOMIC MOLECULE As the simplest example of molecular vibrations, a vibration of a diatomic molecule will now be considered. Being always a linear molecule, a diatomic molecule has only one normal vibration (3 x 2 - 5 = 1). Needless to say, this vibration is a stretching vibration that the molecule stretches and contracts (Fig. 3.2a). We will describe the stretching vibration in accordance with classic mechanics. Assuming that the nuclei are masses, m1 and min, and the chemical bond is the "spring" as k m,

(a)

m2

(b)

Fig. 3.2. (a) A stretching mode of a diatomic molecule. (b) A model for a diatomic molecule (two masses combined by a spring). 23

in Hooke's law (with the spring constant k) (Fig. 3.2b), we can explain the vibration of the molecule based on classic mechanics. Now, assume that the masses m and m2 deviate Ax 1 and Ax2, respectively, from their equilibrium positions. Then, the potential energy of the system shown in Fig. 3.2b is: V = k(Ax 2 -Ax 1 )2

(3.5)

Meanwhile, the kinetic energy of the system is: 1

T =-m~lx

2

-

1

m2 x 2

=

dx('.i>

dt)

(3.6)

Now that V and T are known, motions of the system are determined by solving Lagrange's equation of motion: d(aT + =0 dt (ai ax i

(3.7)

However, before solving Lagrange's equation of motion, we introduce new coordinates Q and X. Q=

pL(Ax 2 -Ax 1 )

(3.8)

+

(3.9)

X = mAx

mAx2

Vm + m2

Now, m2

= m

(where i is a reduced mass)

(3.10)

+M2

Q is a coordinate of the displacement of a distance between the two masses, while X is a coordinate of the displacement of the centre of gravity of the system. Using Q and X, the potential energy V and the kinetic energy T are written as: 24

T

v=

1 Q2 + 1x2

(3.11)

2

(3.12)

2

2

2

k

p

We substitute V and T in the Lagrange equation of motion (3.7). First, applying to the coordinate X (xi = X), we obtain (3.13)

X=0

This expresses a free translational motion which is not bounded by the potential energy. On the other hand, from the Lagrange equation of motion regarding the coordinate Q(xi = Q), we obtain

dQ +k 2 dt

Q

=O

pL

(3.14)

From the differential equation like Eq. (3.14), we can find a solution as follows: Q = Q0 cos2Tvt

(3.15)

Equation (3.15) implies that the system illustrated in Fig. 3.2b has a simple harmonic motion with the frequency v and the amplitude Q0. Substituting Eq. (3.15) in Eq. (3.14), (-42v2 +k)Q=0

(3.16)

Finally, we find the frequency of the spring as: v = I 1k 2n

(3.17)

Since the frequency of the spring corresponds to the frequency of the molecular vibration and the spring constant corresponds to the force 25

TABLE 3.1 Stretching frequencies and force constants of diatomic molecules Molecule

Reduced mass (p) -2 4 (1.66x10 )

Force constants (105 dyne/cm) 1 (N cm )

G(v) (cm - l )

H2

0.50

5.73

4160

HD

0.67

5.77

3631

D2

1.00

5.77

2944

17.50

3.21

556

35C1 2

N2

7.00

22.9

2331

02

8.00

11.8

1555

HF

0.95

9.17

3962

35

0.97

5.16

2886

HBr

0.98

4.06

2558

HI

0.99

3.12

2233

NO

7.46

15.9

1877

CO

6.85

19.0

2143

H C1

constant of the chemical bond, it can be seen from Eq. (3.17) that the frequency of the molecular vibration is proportional to the square root of the force constant and inversely proportional to the square root of the reduced mass of the atoms. Table 3.1 shows the stretching frequencies and the force constants of some diatomic molecules. As can clearly be seen in the table, the stronger a chemical bond is and the smaller the masses of atoms are, the higher the stretching frequency of a molecule becomes. As diatomic molecules, there are those like H2, which consist of the same atoms (homonuclear diatomic molecules) and those like HCl, which consist of different atoms (heteronuclear diatomic molecules). Of these, vibrations of only heteronuclear diatomic molecules appear in infrared absorption spectra. This is because while the electric dipole moment of a heteronuclear diatomic molecule changes with a molecule vibration, that of a homonuclear diatomic molecule is always 0. 26

3.5 QUANTUM MECHANICAL TREATMENT OF A VIBRATION OF A DIATOMIC MOLECULE Quantum mechanics allows us to describe energy levels of vibrations of diatomic molecules. In quantum mechanics, the first step is to write down Schridinger'sequation, HI = Ey. The second step is to solve the equation to calculate an eigen value and an eigen function. In terms of classic mechanics, the total energy H of a vibration of a diatomic molecule is the sum of a kinetic energy 1/2Q 2 (Eq. 3.11) and a potential energy (1/2)k.(Q2/p) (Eq. 3.12), H=T +V =2 Q2 +_

(3.18)

Q2

Replacing Q with an operator -ih/2z.dldQ, H is calculated as: H

d 2 lk -d + -kQ2 8n 2 dQ 2 2 h2

(3.19)

Substituting this in Hy = Ey and processing the formula, a Schrodinger equation on harmonic oscillator of a diatomic molecule is obtained. d2 Y dQ 2

+-

87 2

h2

E-

I

Q2>

k -j)

2

)

=0

(3.20)

As how to solve this equation is described in detail in a number of books, we will explain only a result. The formula (3.20) has a solution only to the following eigen value Ev: E v =(V +-hv

(3.21)

where V is a quantum number of a vibration (V = 0, 1, 2, ...). An eigen function to each value of E v is expressed as: Wv =Nv Hv(,

Q)exp -

(3.22) 27

where Nv denotes a normalization constant, Hv is a Hermite polynomial, and a = 4i72!v/h. Wave functions to V = 0, 1, and 2 are as follows. v/o =(a / 7) 4 exp(-aQ 2 /2)

p1 =(a /

7)14 (2a)V2

,2=(al/t)"

4

(1/

(3.23a)

Q exp(_aQ 2 / 2)

(3.23b)

2)(2aQ2 - 1)exp(-aQ 2 /2)

(3.23c)

As is clearly understood from these formulas, a wave function of a harmonic oscillator is an even function when a quantum number is an even number but is an odd function when a quantum number is an odd number. Figure 3.3 shows a potential energy, wave functions, v, existence probabilities, x' and energy eigen values, E, of the harmonic oscillator. When we treat vibrations of diatomic molecules in accordance with quantum mechanics, we will find different results from when we treat the same in accordance with classic mechanics. Firstly, the lowest vibrational energy is not 0 but Eo = (1/2)hv. Energies have discrete values E = (3/2)hv, E2 = (5/2)hv, E 3 = (7/2)hv, ..., and an energy difference between adjacent energy levels is always hv. Another major (a)

(b)

-Q-0-

+

Fig. 3.3. A potential energy, wave functions, and probabilities of existence of harmonic oscillator.

28

difference between a conclusion we obtain from quantum mechanics and a conclusion from classic mechanics is the amplitude of molecular vibrations. While classic mechanics never allows us to assume that the amplitude, namely, the existence probability of a diatomic molecule extends beyond a potential energy, quantum mechanics permits us to find a slight existence probability outside a potential energy in each state (Fig. 3.3b). 3.6 VIBRATIONS OF POLYATOMIC MOLECULES We will consider normal vibrations of carbon dioxide (CO,; linear triatomic molecule) and water (H 2 0; non-linear triatomic molecule) as examples of the simplest polyatomic molecules. CO2 has 3x3 - 5 = 4 normal vibrations. Figure 3.4 shows the four normal vibrations 1, 3, 2a and 2b (see Section 4.8.1 for labelling rules). The normal vibrations 1 and 2 are vibrations that two CO bonds stretch and contract in phase (1) and out of phase (3), respectively, called symmetric and antisymmetric stretching vibrations. Meanwhile, the vibrations 2a and 2b are both vibrations that the angle of OCO changes and are called bending vibrations.While the vibrations 2a, 2b are independent of each other, energies required for the vibrations are in principle equal to each other, only with planes of the vibrations differing by 90 degrees from each other. That is, the two types of vibrations have the same energy. Such vibrations which have principally the same energy are called degenerate vibrations. 1

3

0

_

2a

2b

0

?

DO

Fig. 3.4. Molecular vibrations of CO2: (1) symmetric stretching vibration; (3) antisymmetric stretching vibration; (2a, 2b) degenerate vibrations. 29

9

1

V,

q

C 2

V 3

Fig. 3.5. Molecular vibrations of water: (1) symmetric stretching vibration (vl); (2) bending vibration (v2); (3) antisymmetric stretching vibration (V3).

To know whether the normal vibrations 1, 3, 2a and 2b are infrared active or not, we examine a change in the electric dipole moment at equilibrium positions (prJQ),. In the normal vibration 1, the electric dipole moment is always 0. Hence, the normal vibration 1 is infrared inactive. Conversely, the electric dipole moment largely changes in the normal vibration 3, and thus it is infrared active. In a similar manner, the normal vibrations 2a and 2b accompany a change in the electric dipole moment, and therefore, are infrared active (see also Section 4.8.5 and Fig. 4.28). With respect to a molecule such as a CO 2 molecule which has the centre of symmetry, a general rule holds true that an infrared active vibration is Raman inactive and a Raman active vibration is infrared inactive. This rule is called the mutual exclusion rule. Water, being a nonlinear triatomic molecule, has three normal vibrations, as shown in Fig. 3.5. The normal vibrations 1 and 3 have different frequencies from each other, because of different H1... H2 interactions between the two (see also Section 4.7.4 and Fig. 4.23). In the cases of CO 2 and H 20 molecules, the frequency of a stretching vibration is higher than that of a bending vibration. This indicates that the stretching vibration requires a higher energy than the bending vibration. Next, let us consider vibrations of atomic groups. Figure 3.6 shows six vibrational modes of an AX 2 group (e.g., CH 2, NH 2). Of the six, two vibration modes are stretching vibrations, one being a symmetric stretching vibration and the other an antisymmetric stretching vibration. The remaining four are bending vibrations, i.e., scissoring, rocking, wagging, and twisting vibrations. Of the four bending vibrations, scissoring and rocking vibrations are bending vibrations in the plane of CH2 (in-plane vibrations), while wagging and twisting vibrations are vibrations which displace vertically to the plane of CH 2 (out-of-plane vibrations). 30

1

2

3

4

5

6

+

T

+

+

Fig. 3.6. Molecular vibrations of AX2 group: (1) symmetric stretching vibration; (2) antisymmetric stretching vibration; (3) scissoring vibration; (4) rocking vibration; (5) wagging vibration; (6) twisting vibration.

In general, group frequencies are useful to consider vibrations of a polyatomic molecule which includes three or more atoms (see Chapter 5). Group frequencies are vibrations of particular atomic groups (functional groups), such as rocking, symmetric and antisymmetric vibrations of a CH2 group, C=O stretching vibration of a carbonyl group and stretching vibration of an OH group. (Bands due to group frequencies are called characteristicabsorption bands.) The concept of group frequencies hold true when certain normal vibrations are substantially determined by movements of two or a plurality of atoms (atomic group). Group frequencies play prominent roles in the analysis of infrared and Raman spectra. A more detailed description of group frequencies is given in Chapter 5.

3.7 QUANTUM MECHANICAL TREATMENT OF VIBRATIONS OF POLYATOMIC MOLECULES We will now explore vibrations of polyatomic molecules in accordance with quantum mechanics. A kinetic energy T and a positional energy V are expressed as: T =2 2 i=1

(3.24) 31

V2 i;iQi

(3.25)

2 i=1

Hence, a total energy H is: 1

n

2V

1

(3.26)

H=T+V= 1Qi +Z-YiQi2 2i 2 i=,

Replacing Q with -ih/2w.d/dQ again and calculating Ht, we can yield a wave equation (3.27) regarding vibrations of polyatomic molecules. h2 -2

87

E

a 2 +i , CqQ2

+

2

-1riQ2 V =EW

(3.27)

Since normal vibrations are independent of each other, the formula above can be separated into n wave equations which, respectively, correspond to the respective normal vibrations, an eigen value E is expressed as the sum of eigen values E i of the respective normal vibrations, and an eigen function yv is expressed as a product of eigen functions yi representing the respective normal vibrations. In short, since the formula (3.27) has the same style as formula (3.20), the eigen value Ei is also the same as formula (3.21). Hence, a total of vibrational energies whose frequencies are v1, v2, ..., Vn is:

(3.28)

E i =(V i +1/2)hv i Ev =E 1 +E 2 + .+En =iV 1 + )hv, +iV 2 +)hv

2

+

+Vn

+2 hv

(3.29)

Figure 3.7 shows energy levels of v1, v2, and v3 modes (Fig. 3.5) of a water molecule. In Fig. 3.7 (0, 0, 0) denotes the lowest ground state and (1, 0, 0), (0, 1, 0), (0, 0, 1) denote fundamental levels at which v1, v2 and v3, respectively, have a quantum number of 1. Transitions between the lowest ground state and the fundamental levels are fundamentals. Next, (2, 0, 0), (0, 2, 0), (0, 0, 2) denote that vj, v2, and v3 have a quantum number of 2, respectively, and are called ouertone levels. (3, 0, 0) ... are 32

(,,

(1 11) 12000

I_______ -(1 _

0 1)

(0 2 1) (O 11) (O 7 8000 0

I--____

_-(1

0

1)

0 0)

(0 2 0) --

(O 1 0) (0 0 0)

4000

A~

Fig. 3.7. Energy levels of vl, vl, and V3 modes of water.

also overtone levels. Overtones are transitions between these overtone levels and the lowest ground state. Combination mode levels are levels, such as (1, 0, 1) and (0, 1, 1), at which two or more normal vibrations are excited. Transitions between combination mode levels and the lowest ground state are called combination modes. 3.8 ANHARMONICITY

So far, we have treated molecular vibrations as harmonic oscillators. However, except for the vicinity of the bottom of a potential energy curve, the harmonic oscillator model is not a good model on molecular vibrations in reality. If the harmonic oscillator model were correct, as we can clearly see in Fig. 3.3, dissociation of molecules should never occur, no matter how large the amplitude is. Hence, it is necessary to consider a potential energy function V(r) (r denotes an inter-nuclear distance) which more accurately expresses vibrations of molecules. In 33

-Q--

o

-

+Q

(re)

Fig. 3.8. Morse's function.

accordance with our instinct, V(r) must be a function such that it rapidly increases when r approaches zero but gradually comes close to a dissociation energy, De, where r >> re (re is an equilibrium distance) holds. As a function which satisfies this condition, Morse's function expressed as below is well known: V(r) = De[1 - exp(-a(r - re))] 2

(3.30)

In formula (3.30), a is a constant. Figure 3.8 shows Morse's function. Assuming that Q = r - re is always smaller than r and expanding V(r) by Taylor's series into a polynomial with respect to Q in the vicinity of re, , Q+ I 2 ) re

24 +-(a8r 4

),

r

~"

IQ+6\ or~(3.31) 6( d3

(3. 31)

Q4 +

Since the first term on the right-hand side is a constant term, this term is regarded as 0. With respect to the second term as well, since V is 34

extremely small to re, the second term is also regarded as 0. Now, ignoring the fourth and higher-order terms and applying (32V ar2 )re = k, the following formula holds: V(r)= 1kQ2 2

(3.32)

In other words, Morse's function is equivalent to a function which expresses harmonic oscillator approximation in the region close to the equilibrium inter-nuclear distance re (a second derivative on formula (3.30) yields k = 2a 2De). A potential energy V is generally expressed as:

V= k2 Q2 + k3Q 3 + k4Q4 + ...

(3.33)

The high-order terms such as Q3 and Q4 are called anharmonic terms. Calculating an eigen value Ev' considering up to the Q3 term, we obtain, EV =V +

hre-IV+

2hvexe

(3.34)

where v, =a / D/ 2 n. The symbol Xe is a constant called an anharmonic constant. It is possible to approximately assume the degree of anharmonicity from the value of this constant. Table 3.2 shows the values of anharmonic constants for major diatomic molecules. While the constant xe has a value of 0.01 approximately, if a molecule contains a hydrogen atom which has a light mass, the constant xe increases (Table 3.2). Since the anharmonic constant xe holds the following relationship with respect to a, De, etc., it is possible to calculate the shape of Morse's function and a dissociation energy of the molecules, etc. ha Xe= hve hv ha 4De 4 2c1*D

(3.35)

From formula (3.34), one can calculate a difference, AEv, between energy levels of vibrational quantum numbers V and V + 1. AE= hv, - 2hvexe(V + 1)

(3.36) 35

TABLE 3.2 Anharmonicity constants of diatomic molecules Molecules

Anharmonicity constant

H2

0.02685

D2

0.02055

HF

0.02176

HC

0.01741

HBr

0.01706

H1

0.01720

C12

0.007081

I2

0.002857

N2

0.006122

02 NO

0.007639 0.007337

Formula (3.36) shows that the larger V is, the smaller AE v is. In this formula, a transition u = 0 -

1 is:

AEv = hve,- 2hvexe = hve(1- 2xe)

(3.37)

Hence, AE = hv, does not hold. The value Vobs (which is an observed value) is obtained as AE v = hvob,. We will now describe a method of calculating ve from Vob.

HCI exhibits a strong band at 2886 cm -1 due to a fundamental (V = 0 to 1) and a weak band at 5668 cm -1 due to a first overtone (V = 0 to 2). It is possible to calculate an absorption wavenumber ve and an anharmonic constant xe from these observed values. With respect to V = 0 AE (1)

= hv( (1-

2

x

)

AE, (,-,) = 2hv,(1 -3x, )

1 and V = 0 -> 2,

(3.38a) (3.38b)

Hence, 2886 = v(1-2x ) 36

(3.39a)

5668 = 2v,(l- 3xe )

(3.39b)

Solving these simultaneous equations, we obtain xe = 0.0174 and ve = 2990 cm - . We must consider Ve to discuss the strength of a chemical bond, since considering Vob, is not sufficient for this purpose.

3.9 OVERTONES AND COMBINATION MODES It is anharmonicity that allows overtones and combination modes to be observed. Let us consider selection rules regarding infrared spectra once again. This time, we will consider anharmonicity on a dipole moment.

I

DQ )o

()

=( 4)Of

Q2 ±.

I Q-L-!

(3.40)

2y Q

edQ+

j J JnQy-dQ (~ Ix m(3.41) =1xlv m +Q

|v

2(,9Q22 )9Q 0dQ+ Q The third term has a value other than 0 when ( 2 pjdQ 2 ) 0 and 2 ynQQ PmdQ X 0 both hold. The latter integral has a value other than 0 when V' V and V + 2. Hence, even a first overtone is no longer forbidden if we consider the term Q2 as well. In a similar manner, second, third, etc. overtones are no longer forbidden as we take higherorder terms into consideration. However, the intensities of these overtones are far weaker than those of fundamentals. The frequencies of first, second, third, etc. overtones are smaller than double, triple, quadruple of the frequencies of fundamentals, respectively. This is because the differences between the vibrational energy levels become narrower as the quantum number u increases, as clearly shown in Fig. 3.9 and indicated by formula (3.36). Anharmonicity excludes combination modes as well from those forbidden in a similar manner. The intensities of combination bands are also weak. 37

3.10 FERMI RESONANCE

In some cases, overtones and combination modes are as strong as fundamentals. This occurs when we have Fermi resonance. Fermi resonance is developed by anharmonicity, when the frequency of an overtone or a combination mode by chance happens to be approximately the same as that of a fundamental (or when an overtone and a combination mode have very close frequencies with each other). When we have Fermi resonance, two relatively strong bands appear in a region where we are supposed to observe only one strong fundamental, one on the high-wavenumber side to the fundamental or the overtone and the other on the low-wavenumber side to the fundamental or the overtone. These two bands both contain contributions from the fundamental and the overtone. In other words, Fermi resonance occurs when the fundamental and the overtone are mixed together because of anharmonicity. Figure 3.9 shows an infrared spectrum of benzaldehyde as an example of Fermi resonance. In general, aldehyde yields a band due to a CH in-plane bending vibration in the vicinity of 1400 cm l. An overtone of this band is expected to appear in the vicinity of 2800 cm l, and its frequency is close to the frequency of CH stretching vibration of aldehyde. In the real spectrum, we find one band on a somewhat higher wavenumber side and another band on the lower wavenumber side to 2800 cm-l (Fig. 3.9). These bands are created because of Fermi resonance between the overtone of the CH in-plane bending vibration and the CH stretching vibration. 2

000 10000

Isd ,,

3200

5T I1 F

I

CUN -

00-__

I- Al~~~J'LL JW L -

2800

(

2400

- -

2000

bU

I&0

/cml

Fig. 3.9. An infrared spectrum of benzaldehyde as an example of Fermi resonance

38

Let us now consider why Fermi resonance occurs. Assume we have small terms (such as a potential energy due to anharmonicity) which provide perturbations to two energy levels, El, E2 (El > E2). We can calculate changes in E1 and E2 caused by the perturbations if we solve the following formula with respect to W. E 1 + al -W

0 =0 E 2 +a

D3

2

(3.42)

-W

where al, a 2, P are the small terms which provide the perturbations. As we expand formula (3.42), we obtain, W 2 -(E

1

+E 2 +

1

2 + a2)W +-(E =0 1 + a,)(E + a2)-P

As we solve this formula (assuming that a, smaller than E1, E2), we obtain, W=E1 +a 1 +

D

E1-

or

E 2 +a E,2

2

E - E2

(3.43)

a 2, p are sufficiently

(3.44)

Changes in E1 and E2 are extremely small if E1 - E2 >> 0. However, when E1 - E 2 = 0, that is, when E1 and E 2 are close to each other, a type of resonance occurs, thereby considerably changing the values of E and E2. When we consider Fermi resonance, we must note that only those having the same vibrational symmetry cause Fermi resonance between them. For instance, although two OH stretching vibrations v1 and V3 (Fig. 3.5) of a water molecule do not cause resonance between them since the vibrations have different symmetries from each other, their overtones (2v1 and 2v3 ) have the same symmetry and therefore cause Fermi resonance (see also Section 4.9.2).

GENERAL READING Theory of molecular vibrations G. Herzberg, MolecularSpectra and Molecular Structure I. Spectra of Diatomic Molecules, 2nd edn. Van Nostrand, Amsterdam, 1950. 39

G. Herzberg. Molecular Spectra and Molecular Structure II: Infrared and Raman Spectra of Polyatomic Molecules. Van Nostrand, Amsterdam, 1945. K. Nakamoto, Infrared and Raman Spectra of Inorganic and Coordination Compounds, 5th edn. Wiley-Interscience, New York, 1997. E.B. Wilson, Jr., J.C. Decius and P.C. Cross, Molecular Vibrations. McGrawHill, New York, 1955. L.A. Woodward, Introduction to the Theory of Molecular Vibrations and VibrationalSpectroscopy. Oxford University Press, London, 1972. Infraredspectroscopy L.J. Bellamy, The Infrared Spectra of Complex Molecules, Vol. 1, 3rd edn. Chapman and Hall, London, 1975; Vol. 2, 2nd edn. Chapman and Hall, London, 1980. N.B. Colthup, L.H. Daly and S.E. Wiberley, Introduction to Infrared and Raman Spectroscopy, 3rd edn. Academic Press, San Diego, CA, 1990. P.R. Griffiths and J.A. deHaseth, Fourier Transform Infrared Spectroscopy. Wiley-Interscience, New York, 1986.

40

Chapter 4

Symmetry of molecules, group theory and its applications in vibrational spectroscopy

In Chapter 3, we learnt that a molecule is infrared active if the dipole moment changes during the vibrational motion and that a molecule is Raman active if the polarizability changes during the vibrational motion. In the case of diatomic molecules, the number of degrees of freedom corresponding to the vibrational motions in the molecule is 3x2 - 5 = 1. This motion is the stretching vibration along the axis of the molecule. We saw in Chapter 3 that the dipole moment of a heteroatomic diatomic molecule changes during its vibration and the molecule is infrared active, and that the dipole moment of a homonuclear diatomic molecule does not change and the molecule is infrared inactive. We then went on to discuss carbon dioxide molecule and identified that the normal vibrations 2, 3a and 3b are infrared active because of the change in dipole moment during these vibrational motions in the molecule. Furthermore, we found that the normal vibration 1 is infrared inactive because the dipole moment does not change during this motion.. However, by the general mutual exclusion principle, as we saw in Chapter 3, we know that this rule holds for molecules with centre of symmetry. That is, the vibrational motions that are infrared active are Raman inactive and the motions that are infrared inactive are Raman active (this must be seen as a rule of thumb not as an absolute principle). To determine whether a motion is Raman active, we must find out whether there is change in the polarizability during this motion. Symmetric vibrational modes in a molecule that lead to a change in the size of the molecule generally involve changes in the polarizability and are Raman active. For example, symmetric stretch41

ing in carbon dioxide leads to a change in the size of the molecule and involves a change in polarizability and the motion is Raman active. For the polyatomic molecules, it is often difficult to determine whether a mode is infrared active or Raman active. The selection rules for infrared and Raman absorption can be arrived at by considering the symmetry of the molecules. In order to do this, we have to learn the symmetry aspects of the molecules, their group and symmetries of the different molecular vibrations.

4.1 SYMMETRY OF MOLECULES: SYMMETRY ELEMENTS AND SYMMETRY OPERATIONS Geometrical figures such as equilateral triangles, squares, cubes etc. possess symmetry. A visual inspection of these shapes leads us to perceive that there are some symmetrical features in them. For example, in a square figure cut out from a white sheet of paper, all the sides are equal. Their diagonals bisect at the centre of the square, the corners of the square are at equal distances from the centre, the square can be divided into two halves along the axis passing through the mid points of any two opposite sides or along their diagonals. Furthermore, the figure can be rotated in several ways about its centre, along the diagonal axes and along the axes passing through the middle points of any two opposite sides to get the same figure without any apparent change. We say that this figure has several symmetry elements. When certain operations are made, the figure seems apparently unchanged. We call these operations symmetry operations. In polyatomic molecules, the orientation of atoms in space may reveal certain symmetry features in the molecules. We can identify various symmetry elements, and symmetry operations that can be performed on molecules. Identifying symmetry elements and understanding symmetry operations are important to classify molecules into different point groups, which we shall consider in the next section. 4.1.1 Identifying symmetry elements and symmetry operations We will take the benzene molecule as an example; it possesses all the symmetry elements we will come across in discussing molecular symmetry and we will try to understand the symmetry elements and 42

2

6

Rotationby60 °

5 2

3

5

C6

4

4

Fig. 4.1. A clockwise rotation of the benzene molecule by an angle of 60 ° (C6) about an axis passing through the centre and normal to the plane of the molecule.

symmetry operations. We mark the corners as 1, 2, 3, 4, 5, and 6 (Fig. 4.1). These numbers will help us to identify any changes in the orientation of the corners representing the carbon atoms in space. Let us assume that the plane of the molecule lies on the plane of the paper. Then we imagine an axis passing through the centre and normal to the plane of the molecule, and consider some rotations of the figure along this axis. A rotation of the molecule (clockwise) about this axis through an angle of 60 ° (2x/6) is performed on the molecule and the new positions of the corners are shown in Fig. 4.1. The molecule is indistinguishable if the numbers indicating the positions of the corners are removed. We can repeat the same operation five more times leaving the molecule apparently unchanged (Fig. 4.2). After the sixth operation, the molecule assumes its original position. The rotation axis is the symmetry element and the rotation about this axis is the operation. This rotation axis is called proper rotation axis (we shall see later that there is defined another rotation axis called improper rotation axis). It is given the symbol C and the order of the axis is written as the suffix. The six-fold proper rotation axis in benzene can be written as C6. The operation generated by this element has also the same symbol. The operations generated by the C6 symmetry element are C6, C62 , C 63, C64 , C65 and C66. As mentioned above, the molecule assumes its original position after the C6 6 operation. This is thus the same as doing nothing to the molecule (C66 = E). The operation is called the identical operation and denoted by E. The molecule also has a three-fold (Fig. 4.3) rotation axis C,, and a two-fold (Fig. 4.4) rotation axis C2, coinciding with the proper rotation axis we identified earlier. The proper rotation axis of highest order is called the principal axis. It is also important to identify equivalent 43

5

1

4

2

(b

4

5

6 4

6 6

3

!

C6

C6

3

5

2

6

(d)

3 6

C6I

4

6

5

5

(a)

4

C6 C6

5

C6

(e)

0

(f)

4

Fig. 4.2. Multiple C6 rotations. (a) Benzene molecule = C6 6; (b) C6; (c) C62; (d) C63; (e) C64 ; (f) C65.

TABLE 4.1 Rotation axes, operations generated and equivalent operations Operation C,

C6

2

C6

3

C6

4

5

C

C

6

6

E C, C2

2

C,

C3 C2

C33 C22

rotations at this stage. Some of the repeated operations generated by the C6 symmetry element may be equal or identical to the symmetry operations generated by the other symmetry elements. For example, the symmetry operations generated by the C6, C3 and C2 can be compared. The operations C62 , C63, C64, C6 6 are equal to operations C3, C2, C32 44

1 6

2 Rotation by 1200

5

3

6

C3

1

(a)

2

4

1

5 6 (c)

Fig. 4.3. C3 operations about the same axis as in Fig. 4.2. 1

Rotation by 180

4

-

C2 2

5

*

6

3 iv lRl

lnttinn

0

1

4 C2

Fig. 4.4. C2 operations about the same axis as in Fig. 4.2.

and E, respectively. Table 4.1 summarises the operations generated by the rotation axes and some of their equivalent operations. The table shows that the distinct operationsgenerated by the C6 rotation axis are C6 and C6 5. The rotation axis C3 generates C3 and C32, and C2 produces only one distinct operation. 45

A

6

2

5

3

-

C2

.

6 6

2

'3

(a) 4

4

SC 2

Cc 2

\

(b)

C2

XC

2

C2 " (c) Fig. 4.5. (a) C2 operations about an axis lying on the plane of the molecule and passing through two opposite corners of the molecule; (b) different C2 as above; and (c) C2 operations about an axes lying on the plane of the molecule and passing through the middle points of two opposite sides of the molecule.

There are other two-fold proper rotation axes lying in the plane of the molecule as shown in Figs. 4.5a, b and c. There are three such axes. We give them the symbols C2 ' and C2 " to differentiate from the two-fold 46

4

1 6

2

5

3

i

3

5

2

6

I 4

t

Fig. 4.6. Inversion operation about the middle point of the benzene molecule.

axis shown in Fig. 4.4. We use the same symbol for all three axes of each type because of our understanding that the operations generated by these axes are equivalent. When specifying symmetry elements, we collect them together. There are six C2 axes lying in the plane of the molecule. As discussed above, it is also clear that the operations C22 generate the identity (C22 = E). Now, we shall consider other symmetry elements in the molecule. Any point on the molecule can be inverted (reflected in the midpoint of the molecule) through the centre of the benzene molecule without any apparent change in the molecule (Fig 4.6). We call this symmetry element inversion centre and the operation inversion. The symbol for the inversion operation is i. The molecule assumes its identical position when operated on twice with i. That is i2 = E. The molecule has several symmetry planes. The symmetry element, symmetry plane generates reflection of the molecule in the plane. The symbol for the symmetry operation is . These planes of symmetry can be differentiated as h, V or Cd. GCh is a horizontal mirror plane lying perpendicular to the principal axis. ov is a vertical mirror plane containing the principal axis which is conventionally taken as vertical. There are three such mirror planes in the benzene molecule (Figs. 4.7a and b). This plane contains the molecular plane of the benzene molecule. d is dihedral mirror plane, a special case of a vertical plane bisecting two C2 axes that lie perpendicular to the principal axis. In the case of benzene there are three dihedral planes, as shown in Fig. 4.7c. Each mirror plane generates reflection and the molecule assumes its identical position when operated on twice with a mirror plane; that is v2 = E, oh2 =

E and

2

Cd =

E.

There is another symmetry element called improperrotation axis or rotation reflection axis. The axis generates a combined operation consisting of an n-fold rotation followed by a horizontal reflection. The 47

cv

Cv

(a)

5

Fig. 4.7. (a) Reflection operation about a vertical mirror plane containing the principal axis and passing through two opposite corners of the molecule; (b) three such vertical mirror planes; (c) three vertical mirror planes containing the principal axis and passing through the middle points of the opposite sides of the molecule; and (d) horizontal mirror plane lying on the plane of the molecule.

benzene molecule has two improper rotation axes S6 and S,3 as shown in Figs. 4.8a and b. Figure 4.8a shows the effect after the first operation. It is important to note that the molecule is in reflected form after the 48

6'

6

I

1'

6

]

i

5

A a

3 3

4 4

I

2' _

.

S6

(a)

1

5

5'

I

6

2

5

3

6'

1'

2

4

2'

S3

(b) Fig. 4.8. Rotation reflection operation.

rotation and indicated by the numbers with primes. The improper rotation axis generates operations S6 , S6 2 , S63, S 64, S65, and 566. A careful study of the resulting molecule after each operation S6 will reveal that some of the above operations are not distinctive. They are equivalent to the operations generated by other symmetry elements of the benzene molecule. The improper rotation axes and their equivalent operations are given in Table 4.2. As in the case of the rotation axes, the distinct operations generated by the improper rotation axis S6 are S, and S65 . The independent operations generated by the S 3 axis are S3 and S3 5. Identifying distinctive operations generated by the symmetry elements is important in our discussion of reducible representations. 49

TABLE 4.2 Improper rotation axes, operations generated and equivalent operations Operation S6

S6

3

4

S6

S62

S6

C3

i

C32

S3 2

S33

S34

h

C3

S,5

S66

E

Operation S,

S,

C3

2

S, 5

S3 6

E

Now it is time to summarise the symmetry elements of the benzene molecule. We have identified the identity E, rotation axes C 6, C3, C2, 3C2', 3C2", mirror planes Gh, 3v, 3 d, inversion centre i and improper rotation axes S6 and S3. Among these symmetry elements, inversion centre and symmetry planes generate one operation each. However, the proper and improper rotation axes generate more than one operation. These operations and their equivalent single operations are given in Tables 4.1 and 4.2. We can now summarise all the symmetry operations that can be generated by these elements as E, C6, C6 5, C3, C32, C2, 3C2', 3C2", i, S 3 , S 35 , S6 , S6 5 oh, 3V and 3 d. We have selected here a molecule with high symmetry and most of the molecules we will be dealing with in this chapter will be simpler than this. TABLE 4.3 Some examples Figure

Symmetry elements

Examples

4.9a 4.9b 4.10a and b

Only E E and a mirror plane E, C2, v' and %c"

CHFC1Br

4.11a and b 4.12a and b 4.13a 4.13b

E, C3, 3ov

NH 3, CHX3, POC13 BF3 , PF,

50

E, C3 , 3C 2,

Gch, 3%v

and S3

CHCl2Br, (CH 3)2CHC1 HO, HS, CH20, COC12, (CH 3 )2C=O, CH 2C12, C6 H5 X, CH 5 N etc.

E, C. and Dcv

HCN

E, C_, ooc and nCh

CO 2

4.1.2 Identifying symmetry elements: some examples Figures 4.9-4.13 illustrate some models of molecules possessing certain common symmetry elements. Table 4.3 summarises the symmetry elements and some examples of molecules possessing these.

(b

Fig. 4.9. Type of molecules (a) possessing E as the only symmetry element, (b) possessing symmetry elements E and a.

0V'-

//

/1

(a)

C2

C2

(b)

Fig. 4.10. Type of molecules possessing symmetry elements E, C2 , %cy'and c%". 51

:YY

Fig. 4.11. Type of molecules possessing symmetry elements E, C3, 3

v.

l

C2

C be

%C

C2

C

Fig. 4.12. Types of molecules possessing symmetry elements E, C3, 3C 2 , 52

Gh,

3%, and S3.

B

infinite number of vertical planes through the principal axis

p Coo-principal axis

(b)

B

-

A I

I

Cw-principal axis

infinite number of vertical planes through the principal axis

(a)

Fig. 4.13. Type of molecules possessing symmetry elements (a) E, C, and cOOv,and (b) E, C,, ooo, and h.

4.1.3 The classification of molecules, point groups The discussion on symmetry elements and inspection of the examples reveal that the molecules containing a different number of atoms may contain the same symmetry elements. For example, the molecules listed in row 3 of the Table 4.3 have the same symmetry elements E, C2, oyv' and ov". The molecules that have the same symmetry elements can be shown to have several properties in common. Therefore, it is advantageous to put all these different molecules into a specific group. The symmetry operations corresponding to the symmetry elements of a molecule leave at least one point invariant (unmoved). We call these groups point groups. 53

-

C,

C--roUDL Fig. 4.14. Example of a molecule belonging to C2 point group.

The molecules are assigned to different point groups according to their symmetry elements. We start with molecules with few symmetry elements and special groups. Molecules of the types shown in Fig. 4.9a possess only the identity element E; they belong to group C. The molecules of the type shown in Fig. 4.9b have E and a vertical mirror plane and they belong to group C,. We see that the molecules of the type shown in Fig. 4.10a and b possess more symmetry elements (E, C 2, ,v' and ov' ) and belong to group C2v. Likewise, the molecules of the type shown in Fig. 4.11 possess E, C3, v,' and v,", v'"' and are said to belong to the point group C3 v. We can clearly see that the classification follows the symmetry elements. Generally, if a molecule has a Cn rotation axis (principal axis) and n vertical planes passing through the principal axis, then it is said to belong to the point group Cnv. Linear molecules of the type shown in Fig. 4.13a have an infiniteorder principal axis and infinite number of vertical planes; the type shown in Fig. 4.13b have, in addition to the above symmetry elements, a mirror plane normal to the principal axis. These belong to the point groups C, and Dih, respectively. Classifying molecules into different point groups does not require the identification of all the symmetry elements. The presence of certain symmetry elements in a molecule implies the presence of certain other symmetry elements. Scheme 4.1 helps us to assign molecules into different point groups (without identifying all the symmetry elements in several cases). 54

no

Scheme 4.1.

TABLE 4.4 Some examples and their point groups Molecules

Symmetry elements

Point group

NH 2 -NH 2 , H2 02

E, C2 (Fig. 4.14)

C2

NH3 , PH3, POC13

E, Ca and 3%7

C3v

Trans CIH=ClH

E, C2 ,

C2h

C2H6 (staggered)

E, C3, 3C2 (horizontal), 3o d (Fig. 4.16)

D3d

C2H2

E, C2 , 2C 2 (horizontal) and ch (Fig. 4.17)

D2h

BF3, PC15

E, C3, 3C 2 (horizontal) and

C6H 6

h

and i (Fig. 4.15)

E, 2C6, 2C3, C2 , 3C2', 3C2", i, 2S3, 2S6 , oh, and

3

D3h

Gh 3

Gd

D6h

oh

55

Oh

C2h- group Fig. 4.15. Example of a molecule belonging to C2h point group.

6C (-U-,

p)-

0

/

(K

C2

f C 2

I

U

D3 d-group

C2

I

C2

Od

Fig. 4.16. Example of a molecule belonging to D3d point group.

4.2 GROUP THEORY AND SYMMETRY OPERATIONS AS ELEMENTS OF A GROUP Mathematically, a set (G) of abstract elements, on which a binary operation o is defined is said to form a group with respect to this operation if the elements of the group obey the following four rules. 1. The product of any two elements A and B and the square of every element is a member of the set. 56

D2 h-group Fig. 4.17. Example of a molecule belonging to D2 h point group.

2. The set contains an identity element E for which EoA = A. 3. The elements follow associative law. That is (AoB)oC = Ao(BoC) 4. For each element A of the set, there exists an inverse in the set such that AoA -1 = E. Mathematically, the binary operation can be defined in several ways. For the purpose of our discussion of symmetry operations, we define this operation as a product operation (one operation followed by another). Furthermore, to avoid confusion with the symmetry elements, we refer the "elements of a group" as "members of the group". Our aim in this section is to show that the set containing symmetry operations as members forms a group. We can make use of the water molecule with four symmetry elements for this purpose. The molecule (Fig. 4.18) has the symmetry elements E, C2, (v') and % (v"). The symmetry operations are also E, C2, oz and o%. We must be aware of the difference between the symmetry elements and symmetry operations. Symmetry operations contain all the individual operations that can be performed about the symmetry elements (see the benzene example). According to the first rule the product of two symmetry operations in the set is a symmetry operation. We shall follow the steps shown in Fig. 4.18 to identify whether the product of the symmetry operations x followed by C2 yields a symmetry operation. It is evident from the illustrations that the product of the above operations leaves the atoms in the molecule in positions that can be transformed by a single operation. This can be simply written as follows. 57

Xz

C2 %yz

Fig. 4.18. Symmetry operations and products of symmetry operations. water molecule is used as an example.

C2(

= oyz

(4.1)

Similarly, from Fig. 4.18 CCC2 = E

(4.2)

A product table (Table 4.5) can be set up to show that this is true for all the operations. This table will also help us to investigate the remaining rules. For example, the following two equations (the multiplication table mentioned above will help here) confirm that the symmetry operations follow associative law. 58

TABLE 4.5 Product table of operations The operation performed first E

C2

aF,2

cy

E

E

C2

a_

1yz

C2

C2

E

ayZ 2

(z

aOxz

a7

ayZ

E

C2

yz

aZ

C2

E

y

(yz C2)xz

=

xz z

=

E

(4.3)

G2 (C2 Gxz) = yz yz = E

(4.4)

The above equations also explain that the elements cv and 2yz have inverses yGxand cy, respectively. This is true for all the symmetry operations in the set. At this stage, it is necessary to mention that the symmetry operations may not be commutative. This means that the result of two successive operations may not be the same if their operation order is reversed. The symmetry operations can be represented by several ways. If there exists a group with other members P, Q, R and S that satisfy the same multiplication table shown above, the group is said to be homomorphous with the group containing symmetry operations.

4.3 MATRIX REPRESENTATION OF THE SYMMETRY OPERATIONS In a Cartesian co-ordinate system the position vector a of a point A can be written as the product of a row vector containing the unit vectors along the co-ordinate axes and a column vector containing the coordinates of the point.

a= (ij k) y

59

If this vector is rotated anti-clockwise by an angle (p about the z axis, the new co-ordinates of the point can be expressed in terms of the original co-ordinates and the angle (p.The z co-ordinate does not change during the rotation. The new x (x') and y (y') co-ordinates can be calculated from the projection of a onto the xy plane. They are x' = x cosq -y sin(p y' = x sinmp + y cosp Z =z

These can be written as a product of a 3x3 matrix and a 3x1 column matrix containing the original co-ordinates. Therefore, the new position vector of the point can be written as the product of a matrix and the original position vector.

x' y'

cosp -sinp 0 x = sin cosp OY0

rx''cos (i jk)y z

=(ij '

-sin(p 0 x k)

sinp 0 O

cosw

O Y 1 z

a' = D(T)a The matrix D(T) is called the transformation matrix (operation) and in this case the transformation is rotation by an angle p. Similarly, symmetry operations and combinations of symmetry operations in a point group can be relatively simplified by turning to matrices. For a summary of matrices and their properties see the appendix. We again use the water molecule Cartesian co-ordinate system as an example. The molecule lies on the XZ plane (a) containing the rotation axis C2 along the Z axis. The H, O and H atoms can then be represented by Cartesian co-ordinates (x1,yi,zl), (x2,y2,z 2) and (x3,y3,z,) 60

-

C2

Oyz

ayZ N

Fxz

Fig. 4.19. Water molecule in three Cartesian co-ordinate systems.

centred at H, O and H atoms respectively. Now we can consider the effect of identity operation (E) on the molecule (Fig. 4.19). Identity operation x1

X1

Y1

Y1

Z, 3C,

Z1

Y2

Y2

Z,

Z2

X,

X3 Y3

X2

Y3 .Z,-

y3 Z,

_z3

_

The operation does not transform any of the co-ordinates and therefore they remain the same. Now let us consider the Cartesian displacements by the operations C2 , az and Gy~ on the molecule. The effects of the operations on the co-ordinates are shown below. 61

C2 -

x1

-X3

X1

rXl

-X 3

Fx1

-y 1 -Yl

Y1 Z1

-Y3

Y1

z3

Z1

X2

-X2

X2

Zl 3C2

x2

-Y2

Y2

-Y2

Y2

Y2 Z,

->

Yz

-I

Y3 Z3

Y2

Z2

Z2

Z2

X3

-xi

X3

X3

-x 1

Y3

-Y1

YZ3

-Y 3

Y1

_Z3

Z3

L

I z2

L Z3

Z3

Z2

Z1

-

If the new co-ordinates after each transformation are represented by a column matrix (X 1,Y 1,Z,X 2,Y 2,Z 2,X 3,Y 3,Z 3), then this column matrix representing each transformation can be written as the product of a 9x9 square matrix containing elements 0, 1 and -1; and the column matrix representing the co-ordinates of the atoms before the transformation.

Matrix representation for identity (E) operation D(E): Matrix A Xi

100000000

xr

Y1

010000000

Y1

Z

001000000

Z1

X2

000100000

X2

Y2

Y2

X3

000010000 000001000 000000100

Y3

z3

000000010 000000001

Z2

_

62

_

_

Z2 X3

_

_

_

Matrix representation for C2 operation D(C2 ): Matrix B 0 0

0 0

0 0 O 0

0 0 000

0

0

Y

0 0

0 0 0 -1 0 0

000

0 0

Z2

0

0

X3

-1

0

0 0

Y3

0

Z3

0

-1 0

0 1

X1

Y1 Z1 X2

0 0 0 0

000 -1 0 0

-1

O 0

x1

-1 0 0 1

Y1

0 0

1 0

0

Z1

0 0

000 000

O 0 0 0 0 0

000

0

X2

Y2 Z2 X3 Y3

0

Z3

Matrix representing oy operation D(3,,): Matrix C X,

1

0

0 0

0 00

Y,

0

-1 0 0

z1

0

0

x1

0 00

0 0

0

Yl

1 0

0 00

0

Z1

1

0 00

0 0 0 0 -1 0

0 0

3C2

0

Y2

0 0

Z2

Xy2

0

z3

00 00

00 00

-1 0 0 0 1 0

00

00

00 00

00 00

0 01 0 00

Y3 X3 Y, 4,

00

0 00

0 1

X3 Y3

Z3

63

Matrix representation for a, operation D(az): Matrix D Xi

0

0 0

0

x1

Y, Z1

0 0

O O 000000 0 0 0 000

1 0 1

Y1

X,

0

O0

-1

0

0

0

0

0

X2

Y2

0

O

0

1 0

0

0 0

Y2

0

O

X3

Z2

0

3

z3

0 -1

0

Z1

0000 1

000

0

0 0

0

0 0

1 0

0

0 0

0

0 0

Y3

1

0

0 0

0

0 0

_Z3

00

-Z'

0

0

-1 0

X3

0

Z2

A group containing the above four matrices A, B, C, D can be shown to obey the multiplication table shown in Table 4.5 and are homomorphous to the C2, point group containing the symmetry operations E, C2, a and a'. The dimension of the matrices representing the symmetry operations depends on the basis selected. In the above case, we have selected Cartesian co-ordinates as the basis and therefore the matrices are 9dimensional. 4.4 CHARACTER OF THE SYMMETRY OPERATIONS The symmetry operations have some symmetry characteristics. These characteristics can be deduced from the characteristics of the matrices representing the symmetry operations. One such characteristic is the character of a matrix representing a particular symmetry operation. The character is the sum of the diagonal elements (trace) in the matrix. For example, the character of the matrix representing the identity operation is 9. The characters of the matrices representing the symmetry operations C2, a(X) and 'a,z)are -1, 3 and 1, respectively. We have used Cartesian co-ordinates as the basis for this representation and therefore the set of characters of the symmetry operations is called the Cartesian representation. This is written as C 2,

E

C0 -

r

9

-1

64

1a 3

1

4.5 CLASSES OF OPERATIONS Two operations P and Q of a symmetry group are said to belong to a class if there is a third operation in the group such that RPR-1 = Q

RR -1 = E

where

We say that Q is the similarity transform of P and that Q and P are conjugate. Conjugate operations fall into the same class. It can also be shown that the operations belonging to a class has the same character. The ammonia molecule belongs to the C3v point group and has the symmetry operations E, C3 , C32, I'v V", "V. By using Fig. 4.20a, we can show that the operations C3, C3 2 belong to a class(Eq. 4.5) and ac'v, v,

Y3

Y

t,I 3

-'2

ACJ

,

-

v

t

?

r,

I

(a) 1

QD' Y

I

I

-1L --

X, C3

£

X,

Y 2

C3 (c)

¥3

Fig. 4.20. A two-dimensional projection of ammonia molecule and symmetry operations. 65

a"'v belong to another class (Eq. 4.6). We know that the inverse of a mirror reflection ('v) is the same reflection ('v). Making the reflection twice leaves the system unchanged. Let us now consider the operation 2 C3 followed by 'v,and C3 . The result is equal to the operation (a"v (Eq.

4.6). G'v C3

o'v = C3 2

(4.5)

The operations C32 and C3 are conjugate and belong to the same class. Similarly, the operations ('v, cy"v and "'v can be shown to belong to another class. C32 'v C3 = ca",

where

C3 is

the inverse of C32 (C32 C3 = E)

(4.6).

Therefore the symmetry operations of the C,3 point group can be simply written as E, 2C3 and 3Gv. We can also work out the characters for the symmetry operations in the Cartesian representations. A twodimensional projection of the molecule is shown in Fig. 4.20. The z-axis is chosen as the principal axis passing through the nitrogen atom perpendicular to the plane of the paper. A C3 rotation will be in the anticlockwise direction for the reader. The effects of operations C3 and a'v are shown in Fig. 4.21. The effects of operations on the co-ordinates and the matrix representations of the operations are shown below. The identity operation E X1

X,

Y1

Y, Z1

Zi X2

X2

Y2

Y2

Z2 X3

Z2

X3

y3

Y3

Z4

X4

z3 y4

z4 _Z,

66

C3

````\

Fig. 4.21. The effect of C3 and Y'v operations on ammonia molecule.

The matrix representation for the identity operation D(E): Matrix A 100000000000 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 000100000000 000010000000

0 0 0

0 0 0 0 0

000001000000 000000100000 000000010000 000000001000 000000000100 000000000010 000000000001 67

The C3 operation X,

-X 3 cos 60 - Y3 cos 30 X3 sin60-Y 3 sin30

Z, X,

Z3 -X, cos 60 Y1,cos 30

Y,

X1 sin60 -Y, sin30

Y2

Z1 -X 2 cos60-Y, cos 30

Z, X,

X2 sin60-Y sin30

Y3

Z3

Z2

X, Yll

-X 4 cos60-X4 cos30 X 4 sin60 -Y4 sin30

z1

Z4

The matrix representation of C3 operation D(C 3 ): Matrix B 0 0

0 0

0 0

0 0

0 0

0 -1/2 -3/2 0 -/3/2 -1/2

0 0

0 0

0 0

0 0

0

0

0

0

0

0

0

0

1

0

0

0

3 /2 0 -1/2 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0

0

-1/2 3/2

68

0

0

1

0

0

0

0

0

0

0

0 0

0 0

0 -1/2 -,3/2 0 0 J3/2 -1/2 0

0 0

0 0

0 0

0 0

0 0

0 0

0

0

0

0

0

1

0

0

0

0

0

0

0

0

0

0 0

0 0

0 0

0 0

00 0

0 0

0

0

0 0

0

0

0

0

0

-01/2 /3/2 -,r3/2 -1/2 0

0 0 1

The c v operation Xi

-- xI

Y,

Y,

Z1

Z1

X2

-X 3

Y2

13

Z2 X3

-x 2

Y3

Y2

z3 X4

-X 4

Z2

y4

Z4

Z4

The matrix representing the v operation is: D(cv): Matrix C --1 00

0

0 0

0

0 0

0

0

0

0 1 0 O 0 1

0 0

0 0 00

0 0

0 0 0 0

0 0

0 0 0 0

0 0 0

O O O O O -1 0 0 O O O 000000 O 1 0 O O O 0000000 O O 1

0 0 0 0 0 0 000

0 0

O O -1 0 0 O O 0 1 0

0 0

0 0 0 0

0 0

0 0 0 0

O O O 0 00 O O O 0 0 0

0 1 00

0 0

0 0 0 0 0 0O 0 -1 0 0

O 000000000 O O O 0 0 0 0 0 0

0 0 0 0

1 0 0 1

69

The characters of the operations in Cartesian representation are C3v,

E

2C3

3c7v

F

12

0

2

4.6 REDUCIBLE AND IRREDUCIBLE REPRESENTATIONS We mentioned earlier that the characters of matrix representations depend on the basis selected. For example, Cartesian representations of molecules belonging to a point group would give different characters depending on the number of atoms. For example, H 2 0 and SO2 molecules give representations consisting of 9x9 matrices each; CH 2 0 a representation consisting of 12x12 matrices; and CH2 C12 a representation consisting of 15x 15 matrices. All these representations formed by these matrices are reducible. To illustrate this, we shall use the matrix representations of the symmetry operations for the water molecule in the C2,, point group using only x coordinate vectors as the basis for the representation. Matrices A, B, C and D representing the symmetry operations E, C2, oy2and z,,will then be (the reader can work this out): Fl

A

0

0 OJ

1 0 001

0 FO C= o 1 0 D= 0 0 0 -1 0

B= 0

-1

0

0- 0

- -

0

-1 0

0

The characters for this representation are 3 -1 3 -1. The above matrices can be similarity transformed using a 3x3 dimensional matrix P such that P- 1AP = A', P-1BP = B', P 1 CP = C', and PDP = D'. The transformation leads to non-zero elements in the leading diagonals of the matrices A', B', C' and D'. The characters of these matrices are the same as the matrices A, B, C and D, respectively (see Appendix III, P). The similarity transformation (reduction) can be continued with A', B', C' and D' until a matrix P is not found to satisfy the above transformations. The resulting matrices A', B', C' and D' will have non-zero diagonal elements. The similarity transformed matrices A', B', C' and D' are: 70

1

(1

1 0 0 A'= 0 1 0

0 1

1 O1 0 B'= 0 -1

10 0 C'= 0 1 0

0

0

0 -1

(1

1)

1

-1

1

1

1 0 D'= 0 -1

0 0

0

-1

0

-1)

The corresponding diagonal elements of the matrices A', B', C' and D' form representations that are called irreducible representations.From the above matrices we have 1 1 1 1, and two 1 -1 1 -1 irreducible representations. 4.6.1 Irreducible representations and character tables A molecule belonging to a point group can have infinite numbers of representations. But all these can be reduced to a combination of a set of irreducible representations. Each point group has a unique set of irreducible representations and they are presented in the character table (see Tables 4.5 and 4.6) of the point group. The character tables (Tables 4.5 and 4.6) need some explanation. The top left corner of the table shows the symbol for the point group. Different classes of operations of the point group are given on the top row in the second column. The number appearing in front of the operations indicates the number of group elements (equivalent operations) in the class. All the operations belonging to a class have the same character. The irreducible representations are labelled by the Mulliken symbols A1, A2, B1 and B2 in the first column of the character table. The following describes their meanings and other symbols we will be encountering later in character tables of different point groups. A collection of character tables is given in Appendix 2. 1. One-dimensional irreducible representations (they have character 1 or -1) are labelled as A or B. The symbol A is used for the irreducible representation that is symmetric with respect to rotation about the principal axis (z-axis that has the highest order of rotation). The symbol B is used if the representation is anti71

TABLE 4.5 Character table of the point group Cz2 C2v

E

C2

A1

1

1

1

1

A2

1

1

-1

-1

B1

1

-1

1

-1

x

RX

xz

B2

1

-1

-1

1

y

Ry

yz

xz

%z X2 y22

Rz

xy

TABLE 4.6 Character table of the C3v point group C3v

E

2Ca

3v

Al

1

1

1

A2

1

1

-1

E

2

-1

0

_ X2 +y2,2

Z Rz (xy)

(Rx,Ry)

(x2 - y 2,xy)(xz,yz)

symmetric with respect to rotation about the principal axis. Twodimensional representations (doubly degenerate) are labelled as E and three-dimensional (triply degenerate) as T. 2. The subscripts 1 and 2 are used respectively depending on whether the representation is symmetric or antisymmetric with respect to a C 2 rotation axis lying in the plane perpendicular to the principal axis (or with respect to a vertical plane if the C2 axis is lacking). Primes (') and double primes (") are used respectively depending on whether the representations are symmetric or antisymmetric with respect to a horizontal plane of symmetry. Subscripts g (geradeeven) and u (ungerade-uneven) are used respectively depending on whether they are symmetric or antisymmetric with respect to centre of inversion (i). 3. The use of subscripts in two-dimensional and three-dimensional irreducible representations also follows certain rules but we will take them as given in character tables. The character table also shows the irreducible representations some directional properties belong to. Their meanings are as follows 72

1. The transformation properties of the rotational modes of a molecule R, R and Rz belong to irreducible representations of the group to which the molecule belongs. These are classified under the respective irreducible representations to which they belong. 2. The co-ordinates x, y and z in a Cartesian system or dipole moment operator (p) or translation operator (T) transform in the same way as the irreducible representations under the symmetry operations of a group. The transformation properties of the Cartesian coordinates under the operations of a group can be easily determined. For example in C2 v, an x co-ordinate under the operations E, C2, , %z transform into x, -x, x and -x (Fig. 4.22). Therefore, the characters of this one-dimensional representation are 1, -1, 1, -1 which are those of the irreducible representation as B 1. Similarly the coordinates y and z can be shown to span the irreducible representations B2 and A. Binary combinations of the co-ordinates (x2, y 2 , z 2, X2 -y 2, xy, xz and yz) are also classified in the same way and placed in a separate column. The degenerate pairs are given in brackets. We shall be using these properties in determining whether a vibration is infrared active or Raman active. Binary combinations can be calculated from the irreducible representations of the co-ordinates x, y and z. For example, in C2v, the characters of the combination x 2 -y 2 (x.x-y.y) can be calculated by using the characters of the co-ordinates x (x spans A with characters 1, 1, 1, 1 for the operations) and y (y spans B 2 with characters 1, -1, -1, 1). The result 1, 1, 1, 1 leads to the classification ofx 2 -y2 under A. 4.6.2 Reducing representations of a point group The process of reducing representations can be achieved by using matrix representations of the operations as shown in Section 4.6. For example, in the case of a water molecule, which belongs to the C2,, point group, the matrices representing the E, C2 , =,and %o,operations can be transformed into another set of representations by using similarity transformations. If the matrices representing these operations are A, B, C and D (as shown above), and if there exists a matrix P and its inverse p-1 such that PAP = A', P-BP = B', P-1 CP = C' and Pr-DP = D' then these can be transformed into a new set of matrices A', B', C' and D' representing the same operations. Their characters remain the same (see appendix on matrices). The transformation is repeated until 73

Z

Z C2 .y

----------

-

~X

x

(a)

.*-------I

-Y

Z Y

Y

i----------

X

(b)

I z XZ z

z Y

I

------

Y

-

-~- ......(c) Fig. 4.22. The transformation properties of the Cartesian co-ordinates under symmetry operations.

all the matrices representing the operations are blocked out as matrices containing non-zero diagonal elements. When such matrices are obtained, each set of corresponding diagonal elements (a'/i, b'ii, c'ii, and d'ii) will belong to one of the irreducible representations of the group. Then the reducible representation can be written as the sum of the irreducible representations. 74

Usually, similarity transformation of matrices involves long and complicated process involving matrices. However, the number of irreducible representations that form a reducible representation can be calculated using the formula below and the character table of the relevant group. Ni =(l/h)

E

NgX(R)Xi(R)

(4.7)

Over all classes

where N = number of times each irreducible representation i appear in the reducible representation; h = order of the group-number of distinguishable symmetry operations in the group; Ng = the number of operations in each class; (R) = the character of the reducible representation for the operation R; Xi(R) = The character of the irreducible representation i for the operation R. We shall illustrate the reduction of the Cartesian representation of water molecule belonging to C2v point group (see Section 4.4). The character table for the point group is shown in Table 4.5. By considering the number of operations in each class (the number shown in front of each operation), we can calculate how many times the irreducible representation A appears in the above reducible representation as follows: N i = (1/h)YNg Xi(R)X(R) 1.E r

9

1.C2

1.a

l.cYyz

-1

3

1

A,

1

N(A1 ) = 1/(1+1+1+1)} [Ng(E)Xi(E)(E) + Ng(C 2) Xi(C Xi(xz) X(xz) + Ng(yz) Xi (Cyz)X( yz)] 5

2 )X(C2 )

+Ng(xz)

= {1/4) [1.1.9 + 1.(-1).1 + 1.3.1 + 1.1.1] = {1/41[9 - 1 + 3 + 1] = 3 75

r

.E

1.C2

1.a_

l.a

i9

-1

3

1

N(A2 ) = 1/4[Ng(E) xi(E)x(E) + Ng(C2 ) Xi(C 2)x(C2) + Ng(Qz) Xi(=) X(o=) + Ng(Oyz) Xi(yZ) X(Oyz) = 1/4[1.1.9 + 1.(1)(-1) + 1.(-1).3 + 1.(-1).1] = 1/4[9 - 1 - 3 - 1] =

N(B 1 ) = {1/4 [Ng() (E) (E) +Ng(C 2) zi(C2)z(C2) +Ng(cz) Xi(o,) &() + Ng(%yz) i(yz) X(%yz)] = {1/41[1.1.9 + 1.(-1).(-1) + 1.1.3 + 1.(-1).1] = t1/4)[9 + 1+ 3 -1] = 3 1.E

1.C2

l.a=

L%,

F

9

-1

3

1

B2

1

-1

-1

1

N(B1 ) = {1/4} [Ng(E) Xi(E)X(E) +Ng(C2) zi(C2)X(C 2) +Ng(5,) Xi(Tzz) X(xz) + Ng,(yz) Xi((Cyz) X(Gyz)] = {1/41[1.1.9 + 1.(-1)(-1) + 1.(-1).3 + 1.1.1] = {1/4}[9 + 1- 3 + 1] = 2 76

These reductions show that the Cartesian representation can be reduced as linear combination of irreducible representations as

F=3A1 +A 2 +3Bl+2B2

(4.8)

Similarly, using the C3Vcharacter table, the reducible Cartesian representation of ammonia molecules can be written as linear combination of the irreducible representations Al, A2 and E.

1.E

Fr

2.C3

12

3.o3

2

A01

1

N(A1 ) = [1/(1+2+3)] [Ng(E) xi(E) X(E) + Ng(C3)Xi(C3) x(C 3) + Ng(ov) Xi(ov) X(Cv)] 1/6)[1.1.12 + 2.1.0 + 3.1.2] = 1/6[12 + 6]

=

=3

1.E 12

r

2.C3 0 11

A2

3.%y 2 -1

N(A 2) = 1/(1+2+3)] [Ng(E)Xi(E)X(E) + Ng(C3)Xi(C3)X(C 3) +Ng(Gv) Xi(Cv)

X(ov)] = 1/6}[1.1.12 + 2.1.0 + 3.(-1).2] = 1/6[12 - 6] =1

77

1.E

2.C3

3.c v

F

i12

0

2

E

2

0

N(B1 ) = {1/(1+2+3)} [Ng(E)Xi(E) x(E) +Ng(C3)xi(C 3)x(C 3) + Ng(ayv) Xi(v) X(Yv)]

= {1/6}[1.2.12 + 2.(-1).0 + 3.0.2] = 1/6[24] =4 Therefore, F=3A,+A2 +4E

(4.9)

The total dimensions of the representations are equal to the number of co-ordinates of the molecule. 4.6.3 Determining characters of operations in Cartesian representations to obtain reducible representations Applications of symmetry to molecular vibrations require: (1) the determination of the group to which the molecule belongs; (2) selection of a suitable basis for the representation of the operations; and (3) determination of the characters of the different operations with respect to the selected basis. The identification of the group to which the molecule belongs was discussed in Section 4.1.3. There are several different ways to select the basis for the representations. However, we will restrict ourselves to Cartesian representations. Determination of the characters of different operations of a group on a molecule is the tricky part which needs some simplification. In the examples shown above for water and ammonia, the use of Cartesian representations resulted in matrices of dimensions 3Nx3N (N = number of atoms in the molecule). Working with matrices of large dimensions is difficult. However, we could follow certain rules that could make the character determination relatively simple: (1) only those atoms that are not shifted during a symmetry operation contribute to the characters (these are the atoms that contribute with 78

non-zero diagonal elements in the matrices; (2) it follows that the identity has a character equal to 3n. Therefore, it is enough to look for atoms that are not shifted during a symmetry operation and sum up the numbers representing the coefficients of the co-ordinates after the operation.

4.7 APPLICATION TO MOLECULAR VIBRATIONS 4.7.1 Vibrational motion, infrared and Raman spectra We learnt in our earlier discussions that vibrational motions of a molecule lead to infrared and Raman spectra of the molecule. The infrared absorption arises when the vibrational motion of a molecule produces an oscillating dipole. The oscillating dipole interacts with the electromagnetic radiation of the same frequency and absorbs it. The infrared spectrum measured by irradiating a sample with polychromatic radiation in the range 4000-250 cm-l contains the absorption patterns of such vibrations. The Raman activity arises from the inelastic scattering of photons from the radiation source by a sample. When a sample is irradiated by monochromatic radiation of visible light, the spectrum measured in the mid-infrared region will contain lines with different intensities. When photons collide with molecules of the samples, some photons lose a part of their energy and are scattered as radiation with less energy (with longer wavelengths). These are called Stokes lines. On the other hand molecules, which are already in an excited state, lose energy and the photons absorb this energy and the scattered radiation will have higher energy and would give lines at shorter wavelengths. These are called anti-Stokes lines. A part of the incident radiation may also pass through the sample without any change in the wavelength. This is called Rayleigh radiation.The infrared region is less energetic than the visible region and Stokes lines are lines with longer wavelengths than the excitation radiation. Therefore, the Raman spectrum measured in the mid-infrared region contains only Stokes lines. When a molecule is exposed to electromagnetic radiation, the electric field component of the radiation distorts the electron distribution in the molecule. This distortion induces a dipole moment that is proportional to the electric field E (Eq. (4.10)). The proportionality constant is called polarizability. 79

p = aE

(4.10)

The polarizability ca is a scalar quantity when pi and E are parallel. Otherwise, it is a tensor quantity with nine components: xx IL= C ay zx

axy

xz

yy ay zy

z

E

( zz,

The polarizability a may oscillate during a vibration and hence the induced dipole moment. The general selection rule for the Raman activity is that the molecule should have an oscillating polarizability.

4.7.2 Vibrational motions and their symmetries We have seen earlier that the internal motions of a molecule containing N atoms can be described by 3N-6 degrees of freedom (or 3N-5 degrees of freedom for linear molecules). These are called fundamental vibrations or normal modes of vibrations or normal modes. Some of these are infrared active, some are Raman active and some are both infrared and Raman active. The Cartesian representation of a molecule provides us with a reducible representation, that can be reduced to a combination of the irreducible representations representing all the modes of the molecule including translation and rotation. In the case of the water molecule, the Cartesian representation is reduced to a linear combination of irreducible representations as F = 3A, +A + 3B + 2B,2 The water molecule has three translational modes and three rotational modes. A look at the character Table 4.5 shows that the translations along the X, Y and Z axes span B, B2 and Al, respectively and rotations about the X, Y and Z axes span B1 , B2 and A 2, respectively. By subtracting these species from the total representation, the modes representing the normal modes of vibrations can be determined. 80

Total representation: Translational modes: Rotational modes:

3A, + A 2 + 3B, + 2B 2 Al + B, + B2 A 2 + B1 + B2

Normal modes:

2A1

+B1

Two of the three normal modes of the water molecule are totally symmetric. 4.7.3 Wavefunctions representing vibrational motion, and their symmetries In quantum mechanics the vibrational states of an electronic state of a molecule are described by wavefunctions. Any vibrational wavefunction of a polyatomic molecule is a product of the wavefunctions of the normal modes. The wavefunction of the excited state of a molecule is also the product of the wavefunctions of all the modes of motion of the molecule. It means that the total wavefunction describing the motion of a molecule with N atoms in an excited state can be written as the product of 3N-6 wavefunctions (or 3N-5 wavefunctions for linear molecules). P = 91929393 ... 93N-6

(4.11)

The wavefunction of the molecule in the lowest vibrational state is similarly composed of 3N-6 wavefunctions. These wavefunctions are of

the gaussian type (= Ce

2

(1 )Y2 ).

The wavefunctions describing the normal modes posses a symmetry related to the symmetry of the normal mode. The wavefunction of a vibrational ground state is totally symmetric (it involves squares of the co-ordinates). In the case of a water molecule, the wavefunction representing the vibrational ground state is of the symmetry type Al. The symmetries of the wavefunctions representing the singly excited vibrational states are the same as the symmetries of the normal modes. Therefore, the symmetries of the wavefunctions of the singly excited vibrational states of the water molecule are A1 , A1 and B2. 4.7.4 Symmetry and infrared absorption When a molecule absorbs infrared radiation at normal temperatures the molecule is excited from the vibrational ground state to the first excited state (Av = ±1 v = 1 - v = 0 and the absorption pattern of the 81

fundamentals falls in the mid IR region. The excitation of a fundamental vibration involves transition dipole moment (transition moment) which is evaluated by the integral (Eq. (4.12)) involving the wavefunctions of the ground and excited vibrational states (pg and (Pe) and the dipole moment 1p of the molecule. The absorption of infrared radiation takes place only if the transition moment has a non-zero value. Transition moment I = p gPp ed

(4.12)

pg is the conjugate wave function of pg. The dipole moment 11 of a molecule arises from the charge distribution in the molecule and can be resolved into three components along x, y and z directions. =

+ y + Pz

PPed =(g(i

I=-

E

JpPi(pdr

(4.13) +iy +plz) ped

(4.14)

(4.15)

i-x,y,x

If one of the above three components has a non-zero value then the normal mode is infrared active. For this to happen, the direct product of the pg, li and (p should span the totally symmetric irreducible representation. Since, the vibrational ground state spans a totally symmetric irreducible representation, the requirement will be met if the direct product between p (ii = x or y or z) and (Pe spans the totally symmetric irreducible representation. In other words i (i = x or y or z) and (Pe must span the same irreducible representation. The dipole moment p is a vector quantity and the components Px, Py and pz span the same irreducible representations as the translation co-ordinates (Fig. 4.22) x, y and z, respectively. This means that if the normal mode spans the same irreducible representation as one of the translational co-ordinates then the mode is infrared active. This is true also for the doubly degenerate (Table 4.7) and triply degenerate irreducible representations because the direct products of all representations with themselves contain the totally symmetric representation. For example, in the case of point group C, the doubly degenerate species has 82

'7 a)

b) Ct V2 = 1595 cm -1

_

= 3651 cm-

Al

Al

,

C) c) V3 = 3755 cm-

B2 Fig. 4.23. Symmetric and asymmetric stretching of water molecule.

TABLE 4.7 Direct product of doubly degenerate irreducible representations

E E ExE = F

E

2C 3

3a

2 2 4

-1 -1 1

0 0 0

characters x(E) = 2, X(C3) = -1 and X(ov) = 0. The direct product of the characters r can be reduced again by using Eq. (4.7) as illustrated in the examples above to r =A 1 +A 2 + E

(4.16) 83

4.7.5 Symmetry and Raman activity An argument similar to the above can again explain the Raman activity in a molecule. The polarizability is a tensor property and is expressed by a matrix containing 9 elements as shown below. The transition moment during absorption can be written as in Eq. (4.15)

I = (P 9

gedr

(4.17)

I = (pgaE(p dT

a; I= ayx (X

(4.18)

a ayy

a

E-

a yz

Ey (PedT

Cy

O z_

Ez

(4.19)

Because of symmetry axy = a,ycayz = azy and a = a,,. For Raman activity, the above integral must be non-zero. This is true if one of the components of the integration is non-zero. An argument similar to infrared activity can be made here to determine whether one of the components is non-zero and hence Raman activity. This means that if one of the components of a spans the same irreducible representation as (Pe then the mode is Raman active. In the above sections, we have simplified the process of identifying whether a normal mode is infrared active or Raman active. They can be summarised as follows: 1. Identify the symmetry group to which the molecule belongs. 2. Develop the Cartesian reducible representation. 3. Reduce the representation as a linear combination of irreducible representations of the group. 4. Identify the irreducible representations spanned by translational and rotational modes. 5. Identify the irreducible representations spanned by the normal modes. 6. Use the character table of the group and decide on whether these normal modes span the same irreducible representation as one of the normal co-ordinates or their product functions and hence whether they are infrared active or Raman active. 84

4.7.6 Measured spectrum and band assignments The above procedure can tell us the number of bands that might arise when the infrared or Raman spectrum of the compound is measured. However, they do not say anything about their assignments in the spectrum. A procedure that could determine the symmetry types of the different absorptions in a measured spectrum would ease the assignments of the bands to different modes. Furthermore, the vibrational modes of a molecule are determined using harmonic consideration of the vibrations. These determinations can be found elsewhere [1]. 4.8 EXAMPLES 4.8.1 Water and nonlinear molecules with the general formulae BAB We have used the water molecule as an example in our earlier discussions. We shall find out whether the three remaining modes of motion representing the vibrations of the water molecule are infrared or Raman or both infrared and Raman active. We found the symmetries of the vibrations to be 2A1 + B2. These are two completely symmetric modes and asymmetric modes as shown in Fig. 4.23. A look at the character table for the C2v point group (Table 4.5) shows that the translational co-ordinates z and y transform in the same way as irreducible representations A, and B2, respectively. Therefore, these modes are infrared active. Furthermore, the product combination of the translational co-ordinates x 2-y2 (and z 2) and yz transform in the same way as the irreducible representations A, and B 2 , respectively. Therefore, the modes are Raman active. All three normal modes of the water molecule are both infrared and Raman active. The bands in the infrared spectrum coincide with the bands in the Raman spectrum. The infrared and Raman spectra of water in a gaseous state contain three bands at 3755, 3651 and 1595 cm -1 . The number of bands is in agreement with our theoretical prediction. However, these do not tell us which of these two bands represents vibrations of A, symmetry. Theoretical calculations show that the Raman scattered light are polarised by the symmetry modes of type A. Therefore, by examining the Raman scattered light with a second polariser, the bands can be classified into different symmetry classes (Fig. 4.23). 85

TABLE 4.8 Fundamental vibrations of water molecule (in gaseous phase) and their characteristics Symmetry type

Label

Assignment

3651

A,(totally symmetric)

VI

Symmetric stretch

1595

Al (totally symmetric)

v2

Bending

3755

B2 (nonsymmetric)

V 3

Asymmetric stretch

Frequency in wavenumber ()/cm

1

TABLE 4.9 The normal modes of some nonlinear molecules with the molecular formulae BAB Symmetric stretch 7l Symmetric bending cml (Al) V2,cm-1 (Al) IR & Raman IR &Raman

Asymmetric stretch V3 cm-1 (B2) IR & Raman

D2O (gas) [2]

2671

1178

2788#

H2S (gas) [3]

2615

1183

2627

H2Se (gas) [4]

2345

1034

2358

NO, (gas) [5]

1318

749

1610

ClO 2 (gas) [6]

943

445

1110

The absorption bands are labelled as v1,v2,v3,

...

vn etc. in the

decreasing frequency order according to their symmetries (see Table 4.8 for water molecule). Note that it is customary in mid-infrared spectrometry to give the frequency in wavenumber (v = cv). The symmetric type bands are labelled first starting from the totally symmetric type. Degenerate vibrations of mode n are labelled as vna, ,,n,etc. The normal modes of some non-linear molecules with a general formula BAB are given in Table 4.9 4.8.2 Ammonia and pyramidal molecules with the general formula AB 3 We can again use the reduced Cartesian representation of ammonia molecule from Eq. (4.9). There are 12 normal modes. The modes represented by reducible representation E are doubly degenerate. The normal modes representing the vibrations can be determined as follows. 86

b)

a) V1 = 3336 cm

1

V2 = 932 cm 1

A,

A1

d)

c) V3a = 3414 cmt E

X

V4,, = 1628 cm

1

E

Fig. 4.24. The normal modes of ammonia molecule.

3A1 + A2 + 4E Total representation: +E Symmetries of translational modes: A A2 + E Symmetries of rotational modes: + 2E 2A1 Symmetries of vibrational modes: The molecule should have 3x4-6 = 6 normal modes of vibration. It is in agreement with the above result where normal mode with symmetry type E is doubly degenerate. The normal modes are shown in Fig. 4.24. A look at the C3v point group shows that z and (x,y) belong to the representations A and E respectively. Therefore, all the four fundamental vibrational modes are infrared active and give four bands in the spectrum (two of these are doubly degenerate). Similarly, the character table confirms that these modes are also Raman active. Here again the bands in the infrared spectrum are 87

TABLE 4.10 The normal modes of some pyramidal molecules and species with formulae AB 3 Antisymmetric Antisymmetric stretch. V,cm-' bend. v4 cm-l

Symmetric stretch. vi cm-'

Symmetric bend. v2 cm-1

(A1 ) IR &

(Al ) IR &

(E) IR &

(E) IR &

Raman

Raman

Raman

Raman

PH 3 (gas) [7]

2327

990,992*

2421

1121

AsH 3 (gas) [7]

2122

906

2185

1005

[C0 3]- (solid) [8]

939

614

971

489

23] (soln.) [9]

967

620

933

469

[S

*Splitting in v2 is due to Fermi resonance (see Section 4.9.2).

coincident with the bands in the Raman spectrum. The normal modes of some pyramidal molecules with general formula AB 3 are given in Table 4.10.

4.8.3 BF3 and planar molecules with formula AX 3 The molecules belong to the D3h point group. The Cartesian representation of the molecule (Fig. 4.25) gives a reducible representation as shown below. The characters were derived with the help of Fig. 4.25. D 3h

E

2C3

3C 2

-

2S3

3ov

Frat

12

0

-2

4

-2

2

The Cartesian representation reduces to a combinationA', + A'2 + 3E' + 2A"2 +E". Of these, translational modes spanA"2 + E' and the rotational modes span A' 2 +E". Therefore, the vibrational modes span A'1 + 2E' + A" 2. These represent 6 normal modes of vibration (3x4-6). The character table for the point group D,, suggests that normal modes with symmetry A" 2 and E' are infrared active because these modes transform the same way as the translational co-ordinates. The normal modes with symmetry A'1 and E' are Raman active. Thus the modes with symmetry E' are both infrared and Raman active. These modes are shown in Fig. 4.26. The normal modes of some planar molecules with a general formula AX3 are given in Table 4.11. 88

C3

Zt

F

Y, C2

0,

F

VI X, ZI

-1

I d

Xl -Y,

II . Fig. 4.25. The effect of symmetry operations of molecules belonging to the D3h point group.

TABLE 4.11 The normal modes of some planar molecules and species with formulae AX 3

BH 3 (gas) [10]

Symmetric stretch. v, cm- 1 (A' l) Raman active

Symmetric bend. v2 cm- 1 (A" 2) IR active

Antisymmetric stretch. V, cm- 1 (E') IR & Raman

Antisymmetric bend. V4 cm(E') IR & Raman

2820

1610

2623

1132

AlBr 3 (gas) [11]

228

107

450-500

A1F 3 (matrix) [12]

660

284

960

93 252

4.8.4 Planar molecules of lower symmetry with four atoms We have discussed the normal modes of pyramidal and planar molecules belonging to the C3 Vand D3h point groups. If one of the atoms 89

t

Ji

( F

Ae_

v1=888 cm

V3a

V4 a

A1

h

v2 =708 cm-l

A2

--

V3 a

3b =1505 cm -

V4 a= V4 b =482 cm

E

t

E

Fig. 4.26. The normal modes of BF3 .

ofX in the planar molecule AX3 is replaced by Y the symmetry elements of the molecule reduce and the molecule will belong to the C2v point group. If Y and Z replace two of the X atoms, the molecule assumes the symmetry elements of the Cs point group. However, the number of normal modes remains constant. Furthermore, the character tables for the C2v and C point groups suggest that all the modes in these molecules are both infrared and Raman active. 4.8.5 Carbon dioxide and linear molecules with the formula XYX The CO2 molecule belongs to the Dh point group. Using the Cartesian co-ordinates a reducible representation of the molecule can be obtained. The characters of the operations for the reducible representation can be determined using Fig. 4.27. The character for the identity operation E is (E) = 9 (all the co-ordinates contribute). When the 90

molecule is rotated about the principal axis by a small arbitrary angle 4, the new co-ordinates X, Y and Z for each atom will be Yi sin4 + Xi cos4, Yj cos4 - Xi sino and Z i (i = 1, 2 and 3), respectively. Each of the atoms will contribute a character of 1+2coso (see Eq. (4.20)) and therefore, (CQ) = 3(1+2coso).Similarly, for reflection on a vertical plane through the principal axis X(av) = 3 (see Fig. 4.27d and Eq. (4.21)). The effect of improper rotation shifts the co-ordinates of the oxygen atoms and, therefore, it is enough to look at the effect on the carbon atom (Fig. 4.27d). The z co-ordinate changes direction and X(S?,)= -1+2coso. The operation inversion moves the co-ordinates of oxygen atoms and therefore, it is enough to look at the character contribution from the carbon atom. All three co-ordinates are reversed and hence x(i) = -3. Similarly, the (ooC 2)= -1. The Cartesian reducible representation is shown in in the table below. Effect of CO on one atom:

cos

sin

0I

-sino

cos 0

0

]0

l Y1

(4.20)

Z

1

Effect of reflection ov on one atom:

cos 2

sin2

Z=[sin2o -cos2

°0

Xl

0

Y

(4.21)

The reducible Cartesian representation is:

Dh

E

2CQ ... Coov

i

2Sj ........ ooC 2

F t

9

3+6cos

-3

-1+2cos

3

-1

Since the molecule has an infinite number of vertical planes and C?, axes along the principal axis (i.e. the principal axis is of infinite order), reducing the above Cartesian representation is difficult. However, a look at the irreducible representations in the character table gives us some clue as to how the above representation can be reduced (see Appendix II for table Dh). In order to obtain a character -1+2cos for the operation 2S?,, the combination must involve 21-, + Hg + - - - . In 91

Zl

C-.0

b)

a)

2 + Ylcos 20

Principal axis

S.o

-.% 'x

Y=Y2 cos

/

C)

-X 2 sin O

Y2

I

Fig. 4.27. The effect of symmetry operations on a carbon-dioxide molecule.

order to get a character +3 for moov operation and -1 for oXC 2 operation, there must be a combination 2 + + Zg+. The total reducible representation reduces to a combination of the irreducible representations as g + JIg + 2u + + 2u. Of these modes, the translational and rotational modes span Yu+ + lu and rIg, respectively. Therefore, the vibrational modes of the carbon dioxide span Xg+ + Eu++ riu. The molecule has four vibrational modes (note that IIu is a doubly degenerate representation) which is in agreement with our prediction (3x3 - 5 = 4). The normal modes of the carbon-dioxide molecule are shown in Fig. 4.28. A look at 92

t

v,= 1340 cm'

yg+

_

-

Vi=134 cm3-

= 2349 cm

(a)

(o 2

(b)

>

-9

2.V-Z

667 cm-'

Fl u

+

+

o~.+

) -

'V2b= 667 cm -' (c)

Fig. 4.28. The normal modes of the carbon-dioxide molecule.

TABLE 4.12 Normal modes of some linear molecules and species with formula XYX Symmetric stretch. v Symmetric bend. v2 cm - (Zg+ ) Raman cm - 1 (Il) IR active active

Antisymmetric stretch v3 cm 1 (Zu+ ) IR active

CS 2 (gas) [13]

658

397

1533

KrF 2 (gas) [14]

449

233

596, 580

[CuCl 2]-(s) [15]

300

109

405

the character table for point group Dh shows that the translational coordinates span u,+ and Iu. The products x2 + y 2 and z 2 span g+. That is, the anti-symmetric stretching mode and degenerate bending mode are infrared active and the symmetric mode is infrared inactive but Raman active. This example also illustrates the mutual exclusion principle for molecules with a centre of inversion. The normal modes which are infrared active are Raman inactive and those which are Raman active are infrared inactive. The normal modes of some linear molecular formula XYX are given in Table 4.12. 93

4.8.6 Linear molecules with formula XYZ When one of the atoms ofX is replaced by another atom Z, the molecule assumes symmetry elements of the Cv point group. The character table indicates that all the three normal modes are both infrared and Raman active.

4.9 MEASURED VIBRATIONAL SPECTRA OF MOLECULES We have learnt in this chapter how to predict the vibrational spectra of molecules. However, measured spectra often show more bands than the predicted 3N-6 or 3N-5 fundamental frequencies and, in some cases, there are fundamental frequencies that cannot be observed in the infrared spectrum.

4.9.1 More bands due to anharmonicity-overtones and combination bands One of the reasons for observing more bands than expected arises due to anharmonicity of the normal vibrations. The normal vibrations of a molecule were assumed to be simple harmonic. However, this is not the case. The bonds between the atoms behave like anharmonic oscillators and transitions between vibrational ground state and higher levels become possible. The selection rule for anharmonic oscillators is Av = +1, +2, +3, .... The transition between vibrational ground state to the second excited state (v = 2 - v = 0) is called first overtone (second harmonic) and to the third excited state (v = 3 - v = 0) is called second overtone (third harmonic). The frequencies of these transitions are not exact multiples of the fundamental transitions. The frequencies of o) < 2 V_ these transitions decrease (i.e., V=2There is also another possibility for the transition to occur to a combined level. This can be the sum of the tones such as v, + v2 , v1 + v2 + V3 , etc. or 2v, + v2 , v1 + 2 V2, etc. or difference tones such as v¾ - v2, v2 - V,, etc. These transitions normally fall in the near infrared region of the electromagnetic spectrum. However, some of them can be observed in the mid-infrared region. In such cases the number of bands in the mid-infrared region exceeds the number of bands predicted by group theory. 94

4.9.2 More bands due to Fermi resonance Two molecular vibrations may interact with each other if they have frequencies very close to each other (30 cm-'). For example, one of the fundamental modes and an overtone of another mode or a combination mode may have frequencies close to each other (accidentaldegeneracy). These vibrations may interact if their symmetries are the same and the overtone or combined tone is enhanced. The interacting vibrations split and the vibration with higher frequency is raised in frequency and the vibration with lower frequency is depressed about the average of the two vibrations. This is called Fermi resonance (see Section 3.10 for treatment of Fermi resonance). An example of Fermi resonance is the interaction between the symmetric stretch (I) around 1340 cm - ' and the first overtone (2v2 1334 cm-') of the symmetric bending at 667 cm - in a carbon dioxide molecule. The symmetric stretch has symmetry g+. The symmetric bending is a doubly degenerate vibration with symmetry Ig. The first overtone of this degenerate vibration splits into two sublevels with symmetry species g+ and Ag. Fermi resonance arises because of the interaction between the species Xg+ of the symmetric stretching vibration and the species Eg* of the first overtone (of the symmetric bending vibration). This interaction results in two bands in the Raman spectrum at 1388 cm -1 (the frequency is raised about the average 1340 cm-1 ) and 1286 cm 1 (the frequency is lowered about the average 1340 cm-1 ). 4.9.3 Overlap of rotational spectrum on vibrational spectrum The rotational spectrum of some small molecules (diatomic and triatomic) falls into the same region as the vibrational spectrum of the molecule. The fine structure of the rotational spectrum can be seen in the infrared of the molecule. For example, the infrared spectra of carbon monoxide (see Fig 2.5) and the ammonia molecule are accompanied by their rotational spectra. However, this is not a problem in polyatomic molecules because the rotational bands are very close to each other and hardly visible in the spectra. REFERENCES 1. 2.

G. Herzberg, Infrared and Raman Spectra. Van Nostrand, New York, 1945. W.S. Benedict, N. Gailar and P.K. Plyler, J. Chem. Phys., 24 (1956) 1139. 95

3. H.C. Allan and P.K. Plyler, J. Chem. Phys., 25 (1956) 1132. 4. D.M. Cameron, W.C. Sears and H.H. Nielsen, J. Chem. Phys., 7 (1939) 994. 5. R.V. St. Louis and B.L. Crawford, Jr., J. Chem. Phys., 42 (1965) 857. 6. A.H. Nielsen and P.J.H. Woltz, J. Chem. Phys., 20 (1952) 1878. 7. E. Lee and C.K. Wu, Trans. FaradaySoc., 35 (1939) 1366. 8. W. Sterzel and W.D. Schnee, Z. Anorg. Allg. Chem., 383 (1971) 231. 9. J.C. Evans and H.J. Bernstein, Can. J. Chem., 33 (1955) 1270. 10. A. Kaldor and R.F. Porter, J. Am. Chem. Soc., 93 (1971) 2140. 11. I.R. Beattie and J.R. Horder, J. Chem. Soc., A (1969) 2655. 12. A. Snelson, J. Phys. Chem., 71 (1967) 3202. 13. T. Wentink, J. Chem. Phys., 29 (1958) 188. 14. H.H. Classen, G.L. Goodman, J.C. Malm and F. Screiner, J. Chem. Phys., 42 (1965) 1229. 15. D.N. Waters and B. Basak, J. Chem. Soc., A (1971) 2733.

GENERAL BIBLIOGRAPHY P.W. Atkins, Molecular Quantum Mechanics. Oxford University Press, London, 1984. C.N. Banwell and E.M. McCash, Fundamentals of Molecular Spectroscopy. McGraw-Hill, London, 1994. M. Ladd, Symmetry and Group Theory in Chemistry. Horwood Publishing, Chichester, 1998. K. Nakamoto, Infrared and Raman Spectra of Inorganic and Co-ordination Compounds. Wiley, New York, 1978. J.D. Ronaldson and S.D. Ross, Symmetry and Stereo Chemistry. Intertext Books, London, 1972 A. Vincent, Molecular Symmetry and Group Theory. Wiley, London, 1977.

96

Chapter 5

Group frequencies and assignments of the infrared bands

5.1 GROUP FREQUENCIES Analysis of normal vibrations of a polyatomic molecule is, in general, complicated because the molecule consists of a number of atoms. Although the vibrations of a polyatomic molecule can be calculated by calculating a potential energy and a kinetic energy of a system, as in the case of a diatomic molecule, the calculation is not straightforward. The concept ofgroup frequencies helps the analysis of vibration spectra of a polyatomic molecule. The concept of group frequencies may be applicable when the amplitudes of nuclei in a particular functional group are very large in a certain normal vibration, while those of the other atoms are very small. Infrared spectroscopy is useful for qualitative and quantitative analysis and structural investigation of a complex molecule since there are a number of vibrational modes which can be regarded as group frequencies. Table 5.1 summarizes group frequencies. Group frequencies observed mainly in Raman spectra (e.g., S-S stretching vibration) are also shown in this table. The idea of group frequencies is beneficial even for a very complex molecule such as protein. As an example, let us consider normal vibrations of an amide group. Normal vibrations of an amide group have been calculated in detail, taking N-methylacetamide (Fig. 5.1) as a model of the amide group [1,2]. Considering a methyl group as one atom, N-methylacetamide is a six-atom molecule, and hence, has twelve normal vibrations (3x6 - 6 = 12). Of the twelve, the normal vibrations shown in Fig. 5.1 are amides I, II and III modes which are key vibrations for studying the structure of proteins. As can clearly be seen in Fig. 5.1, amide I has a strong characteristic of C=O stretching 97

TABLE 5.1 Group frequencies 3500

3000 I OH stret NH stret

2500

2000

1500

1000

500

T

I Casym bend · CI sissor

CH stret(unsaturated)

CIH sym bend

CHstret(saturated)

CHl:wag _ CHbend I JC= C)

m SH start

* 01 str tn IC

P=O- stret

stret CLO stret

I PO; antisym sret I PO;sym stret

* C=Ostret(COOI) * -CONH-

i

C=Cstret

* C=-Nstret _ CO, antisym stret I NO antisym street

*N=N stret I -CONH-

* N-N stret _ CO2 sym stret NN atisym stret NO2 sym stret °(c:Na

~

Rr.mtgd(p

htA

rmm~ri~ ~ I

I

3500

3000

C-N stret C-C street

~C-O

street Sis

2500

2000

1500

1000

500

Wavenumber Icni' vibration. Meanwhile, amides II and III are coupling modes of C-N stretching vibrations and N-H in-plane bending vibrations. Of the three, amides I and II appear strongly in infrared spectra, and amides I and III appear intense in Raman spectra. Amides I, II and III bands of C\, 3

H

C-N 0t

\CH3

Fig. 5.1. Structure ofN-methylacetamide and its amide I, amide II, and amide III modes. (Reproduced from Ref. [2] with permission. Copyright (1984) Academic Press.) 98

TABLE 5.2 Secondary structures of protein and frequencies of amide I, II, and III bands Secondary structures

Infrared

Raman

Amide I

Amide II

Amide I

Amide III

a-helix

1655-1650

3-sheet structure

-1690, 1680-1675, -1640, -1630

-1540

1660-1645

1300-1265

-1550

1675-1665

1240-1230

Random coil

1655-1645

1535-1530

P-turna

-1680, -1660, -1640

310helixa

1645-1640

31helixa

-1640

1670-1660

1260-1240

-1680 ,-1660, -1640

1330-1290

-1655 -1550

-1380, -1335, -1285

aFrom Ref. [4].

proteins are generally found in the range of 1690-1620 cm - l , 1580-1520 cm l and 1320-1220 cm -1 , respectively. The frequencies of these modes are known to sensitively reflect secondary structures of polyaminoacids, peptides, and proteins [3-6]. Table 5.2 shows relationships between secondary structures of proteins and the frequencies of amides I, II and III. Measurement of infrared (or Raman) spectra of protein allows us to estimate the contents of secondary structures, which will be explained in Section 8.7.1.

5.2 ISOTOPE SHIFT In analysis of infrared spectra, a shift associated with isotopic substitution (isotope shift) gives solid assignment of bands in many cases. In general, force constants may be assumed not to change due to isotopic substitution, and therefore, isotope shifts lead to mass effects alone. Calculate the magnitude of an isotope shift, taking a diatomic molecule as an example. The frequency of a stretching vibration of a diatomic molecule is given by Eq. (3.17). Where v' is the frequency for replacing an atom having a mass ml with an isotope having a mass ml', the following relation holds: 99

Fig. 5.2. Amide I', amide II', and amide III' modes of deuterium-substituted N-methylacetamide. (Reproduced from Ref. [2] with permission. Copyright (1984) Academic Press.)

v=

~

(5.1)

p' = m,' m 2/(m' + m2) holds. As Eq. (5.1) clearly shows, the larger the difference between ml and m,', the larger is the isotope shift. Since v/v' = 1.36 if H is replaced with D, a C-H stretching vibration of saturated hydrocarbons, which appears in the vicinity of 2900 cm l, shifts close to 2100 cm-l. Figure 5.2 shows amides I, II and III modes of deuteriumsubstituted (ND) N-methylacetamide (which will be called amides I', II' and III'). While shifts induced by the deuterium substitution are rather large in the amide II and III modes as NH bending vibrations contribute to these two modes, amide I mode, being principally a C=O stretching vibration, gives rise to a very small isotope shift. What should be noted with respect to isotope shifts of polyatomic molecules is that vibrational modes change more or less in association with isotopic substitution, which can be clearly understood from comparison of Fig. 5.1 with Fig. 5.2. In studies of infrared spectra, 15N-substitution, 13Csubstitution and the like are often utilized in addition to deuterium substitution. Although an isotope shift is small when such a heavy atom is replaced, a change in a vibrational mode associated with the isotopic substitution is also small. 5.3 HOW TO MAKE BAND ASSIGNMENTS IN INFRARED SPECTRA In general, we observe a number of bands in an infrared spectrum of a molecule. The first consideration to be made when analysing the infrared spectrum is whether it is necessary to analyze the infrared spectrum as a whole or only a part of the spectrum. For example, if an 100

objective is to examine a secondary structure of protein, the spectral regions where the amide I and amide II bands appear may be studied and it is not necessary to analyze the entire spectrum. On the other hand, when we want to identify a certain material, we must analyze a considerable portion of the spectrum. Although there is no absolute procedure for assignments of infrared bands, the following methods are often found effective: 1. Find group frequencies from comparison of observed frequencies with a table of group frequencies. During the comparison, intensities as well as frequencies must be noted. 2. Compare an obtained infrared spectrum with those of similar molecules. The similar molecules do not always need to be similar in their entirety, but rather, may be only partially similar. 3. Measure an infrared spectrum of an isotope-substituted material which contains deuterium, 15N, 13C, etc., and compare the infrared spectrum with an original spectrum. This method is very effective for the identification of a band due to a particular functional group, a particular bond, etc. 4. Measure spectra while varying a condition of a material, such as a temperature, pH and a solvent, and compare them with an original spectrum. For example, since amino acid residues within protein, respectively, have unique pK values, as pH is changed around the pKvalues, it is possible that only bands due to particular amino acid residues will change. 5. Measure anisotropy of infrared absorption resulting from polarized light. This method is convenient to distinguish in-plane and out-ofplane vibrations of a planar molecule from each other. 6. Calculate normal vibrations. Although this is a traditional method, if not combined with another scheme, it does not allow us to completely assign an infrared spectrum of a complicated molecule. This is because of a general lack of knowledge regarding threedimensional structures and force constants of molecules. 7. Try chemometrics and two-dimensional correlation analysis (see Chapter 9) There are unique marker bands known, some unique to proteins, some unique to nucleus acids, etc., with which we can study structures of 101

materials (amides I and II are typical examples of marker bands for proteins). Hence, for actual analysis of an infrared spectrum of a molecule, it is often important to first identify marker bands. Marker bands for organic thin films, polymers, and biological molecules and the like will be described in Chapter 8. A library search alone is often sufficient if the objective is merely identification of a material based on measurement of infrared spectra. It is very convenient to know which bands appear intense in an infrared spectrum for its analysis. The well-known principles regarding intensities of infrared bands are as follows: 1. A band due to a functional group with a strong polarity appears strongly, e.g., OH and C=O stretching vibrations. Conversely, vibrations due to a bond with a weak polarity, such as C-C and S-S stretching vibrations, appear very weakly or do not appear in an infrared spectrum. 2. Antisymmetric stretching vibrations are stronger than corresponding symmetric stretching vibrations. For example, COOantisymmetric stretching vibrations appear more strongly than COO- symmetric stretching vibrations. 3. As a general tendency, local vibrations are strong and vibrations of a molecule as a whole are weak. For instance, in an infrared spectrum of polyethylene, while CH 2 rocking vibrations appear strongly, vibrations in which a molecule as a whole stretches and contracts (e.g. accordion vibrations) appear weakly. 4. Among bands arising from an aromatic group, those which give strong infrared bands are ring stretching vibrations in the 1600-1450 cm 1 region and out-of-plane bending vibrations in the 900-700 cm 1 region. REFERENCES 1. 2. 3. 4. 5. 6.

102

T. Miyazawa and E.R. Blout, J. Am. Chem. Soc., 83 (1961) 712. Y. Sugawara, A.Y. Hirakawa and M. Tsuboi, J. Mol. Spectrosc., 108 (1984) 206. H. Susi and D.M. Byler, Arch. Biochem. Biophys., 258 (1987) 465. S. Krimm, in: T.G. Spiro (ed.), Biological Applications of Raman Spectroscopy, Vol. 1. Wiley, New York, 1987, p.l. M. Jackson and H.H. Mantsch, CRC Crit. Rev. Biochem. Mol. Biol., 30 (1995) 95. H.H. Mantsch and D. Chapman (eds.), Infrared Spectroscopy of Biomolecules. Wiley-Liss, New York, 1996.

GENERAL BIBLIOGRAPHY L.J. Bellamy, The Infrared Spectra of Complex Molecules, Vol. 1, 3rd edn. Chapman and Hall, London, 1975; Vol. 2, 2nd edn. Chapman and Hall, London, 1980. N.B. Colthup, L.H. Daly and S.E. Wiberley, Introduction to Infrared and Raman Spectroscopy, 3rd edn. Academic Press, San Diego, CA, 1990. E. Maslowsky, Jr., VibrationalSpectra of OrganometallicCompounds. Wiley, New York, 1976. K. Nakamoto, Infrared and Raman Spectra of Inorganic and Coordination Compounds, 5th edn. Wiley, New York, 1997. N.B. Nyquist, The Interpretation of Vapor-Phase Infrared Spectra Group Frequency Data. Sadtler Research Laboratories, Philadelphia, 1984. G. Socrates, Infrared Characteristics Group Frequencies. Wiley, New York, 1980.

103

Chapter 6

Instrumentation

6.1 HISTORY OF INFRARED INSTRUMENTATION Since the discovery of infrared radiation by Sir William Herschel in 1800 [1], a variety of methods have been used to improve the experimental techniques for measuring infrared spectra. In particular, Herschel used a prism and a mercury thermometer to record his observations of heat-based radiation beyond the range of the solar spectrum. Melloni is credited with the construction of the first midinfrared spectrometer in 1833, after his discovery of the transparency of NaCl in the infrared. Excellent references on the early history of vibrational spectroscopy and the minds that shaped the field can be found elsewhere. This chapter will deal in detail with the interferometric methods of capturing infrared radiation. Older monographs treat non-interferometric methods in great detail [2]. 6.1.1 History of FT-IR instrumentation The construction of the first interferometers dates back to 1880. Lord Rayleigh may have constructed an interferometer at that period, but it is Michelson who is credited with the construction of the first operable instrument at that time. The interferometer was used in the measurement of the speed of light in diverse directions. Measurements that took place at was is now Case Western Reserve University showed that there was no detectable difference in the speed of light in either direction. The lack of adequate computing power was the main reason that it took approximately eighty years for the instrument to utilize its full potential as an analytical tool. Ferraro has recently published a very informative account of the history of FT-IR spectroscopy [3]. 105

The first attempts to use interferometric means to measure infrared radiation concentrated in the far infrared region of the spectrum. The operational needs for mechanical precision and computational load are lower in the far-infrared compared to the mid-infrared. Therefore, a lot of the early applications and advancements took place in the field of astronomy where far-infrared spectroscopy is used extensively. One very important recent development that led to the widespread use of infrared spectroscopy as a characterization tool was the introduction of the first Fourier transform infrared spectrometer, the FTS 14, by the Biorad Company of Cambridge, Massachusetts in 1969. Several developments in the 1950s and 1960s contributed to the introduction of this first commercially available instrument. The work of L. Mertz in interferometer design during 1954-56 and the development of data reduction algorithms in 1960-65 are probably the most significant contributors. One historical aspect that should not be overlooked was the role of the NASA contract to Block Engineering for an instrument with ten times the available resolution. The model 1500 (296 in the commercial version) had 0.5 cm-l resolution and a better signal-tonoise (S/N) ratio than the dispersive instruments of the time. Other notable developments were the discovery of the HeNe laser along with the introduction of better infrared detectors, analog-todigital (A/D) converters and minicomputers. In 1966 a one-foot laser with a built-in power supply was available to be used in the first FT-IR spectrometer. In addition, pyroelectric bolometers in the form of the deuterated triglycine sulphate (DTGS) detector also became available. Their major advantage was that their bandwidth was compatible with the rapid-scan frequencies. With respect to computing power there was a remarkable development from the PDP-1 in 1960 to the DG Nova in 1969. After the introduction of this instrument many new instruments from various manufacturers have appeared and tremendous progress has been achieved in the years following 1969. A list of the advancements will undoubtedly include the very small footprint of the modern instruments, quadrature detection with forward and backward scanning, digital signal processing, diagnostic features, low powered aircooled sources, the flexible design of the research-grade instruments, the multiple spectral ranges, the very high spectral resolution, and the tremendous progress in FT-IR software. In addition, one of the most important developments was the 'rediscovery' of step-scan interferometry, a subject that will be extensively dealt with in this book. 106

What is the future of infrared instrumentation? Without a doubt, any technological breakthrough will eventually find its way to a commercial design with time. Improvements in performance, new features and capabilities, usability and (hopefully!) a reduction in cost should all be expected.

6.2 COMPONENTS OF AN FT-IR SPECTROMETER The components of an FT-IR spectrometer primarily include the source interferometer, the source of radiation, the detector and other optical elements (beamsplitters, mirrors, etc.). In addition, data manipulation also takes place in the adjacent computer station. It is beyond the scope of this book to give a detailed account of all the elements involved; instead, an attempt will be made to cover in more detail important components of these designs. 6.2.1 Sources The source of infrared energy in an infrared instrument does not depend on the type of instrumentation used to detect the radiation. Both dispersive instruments and Fourier transformed instruments can use the same types of infrared sources. Therefore, a general overview of the available technology will be reviewed here. The more typical sources are the Globar source and the Nernst glower, even though nichrome coils have also been used in the past. Nichrome coils operate at lower temperatures and therefore have a lower emissivity. Finally, mercury arc lamps are used most frequently for experiments dealing with the far-infrared region of the spectrum. The first two types will be discussed in more detail here. GlobarSource This source is made out of silicon carbide (SiC) and it has metallic leads at the ends which serve as electrodes. The application of electric current results in the generation of heat, which yields radiation at temperatures higher than 1000°C. Water cooling is required for this type of source because the electrodes need to be cooled [4]. This extra level of complexity makes this source less convenient to use and more expensive. Figure 6.1 shows the ratio of the globar source to a 900°C blackbody. 107

Prisms: 4.2. 4.2.

.

NaCI & KBr 0

Ramsey and Alishouse

Q

3.4

s

CsI

Calculated after Silverman

1.8

1.0 0.2

I I i

2.l

6.0 10.0 14.0 18.0 22.0 26.0 30.0 34.0 38.0 Wavelength (m)

Fig. 6.1. Globar versus a 900°C blackbody. (Reproduced from Ref. [7] with permission. Copyright 1968, Pergamon.) I.o

0.9 W 0.8 E 0.7 K 0.6

-255 K Stewart and Richmond

0.

n_

I

1

II 3

I I

5

I I I I 7 9 11 Wavelength (im)

I

I 13

I 15

Fig. 6.2. Spectral emissivity of the globar source. (Reproduced from Ref. [7] with permission. Copyright 1968, Pergamon.)

One other advantage of this source is its high emissivity down to 80 cm - ', making it useful in the far-infrared region of the spectrum. Figure 6.2 shows the spectral emissivity of a typical globar source [5]. These values are only representative and are expected to change considerably with use. Recently, a new low power air-cooled ceramic source has been introduced into the modern FT-IR instruments [6]. This source has the advantage that no watering cooling is necessary, making their instruments portable and easier to maintain. Nernst glowers This infrared source's element is a mixture of yttrium and zirconium oxides and has an emission spectrum that resembles that of a black 108

z

}

3 W

Wavelengt

(m)

Fig. 6.3. Ratio of a Nernst glower to a 900°C blackbody. (Reproduced from Ref. [8] with permission. Copyright 1978, Office of Naval Research.)

body at 1800 K. It is an insulator at room temperature and becomes a conductor after it is preheated. It used to be popular but it has a number of disadvantages, the biggest being its short lifetime and mechanical instability. The spectral characteristics of a Nernst glower versus a 900°C blackbody can be found in Fig. 6.3 [7]. 6.2.2 Infrared detectors One of the most important elements of an infrared spectrometer is the component responsible for the detection of infrared energy. Typically, the description of a detector is not limited to the responsive element which changes the incoming radiation into an electrical signal, but it also includes the physical mounting of the element, like the windows, the apertures, the Dewar flasks, etc. Together, they form what is called a detector [8]. There are two general classes of infrared detectors. One class comprises the thermal detectors and the other class the photon detectors. As the name suggests, thermal detectors operate by sensing fluctuations in the temperature of an absorbing material as a result of exposure to the incoming radiation. The other category, the photon or as often called quantum detectors are sensitive to changes in the quantity of free-charged carriers in the solid, brought by the interaction with the external radiation. 109

Thermal detectors Thermal detectors rely on four different processes to achieve detection of infrared radiation: 1. The bolometric effect. This effect relies on the change in the electrical resistance of the responsive element due to temperature changes produced by the absorbed infrared radiation. This change in resistance is detected by conventional techniques. 2. The thermovoltaic effect. In this case, the heating of the junction between two dissimilar materials produces a measurable voltage across the leads. 3. The thermopneumatic effect. A very common thermal detector, the Golay detector, relies on this phenomenon [9]. In the case of thermopneumatic detectors, a gas-filled chamber that contains an infrared absorbing element is exposed to infrared light. Absorption of energy by the element generates heat, which heats up the gas in the chamber. The consequent increase in the pressure of the gas results in the distortion of a thin flexible mirror on the other end of the sealed gas chamber. This distortion is sensed by an independent optical system. Golay detectors have been extensively used as farinfrared detectors, even though they had problems with their mechanical integrity at one point. Figure 6.4 shows a cross-section of a typical Golay cell [10]. 4. The pyroelectric effect. In this process the radiation increases the temperature of a crystalline material. The result is a change in the electrical polarization of the crystal surface and the generation of an electric field. TARGET: THIN

BSORBING FILM IR TRANSMITTIN WINDOW

ELASTIC MEMBRANE

GAS FILLED CHAMBER Fig. 6.4. Cross-section of the Golay cell. (Reproduced from Ref. [10] with permission. Copyright 1976, Institute of Optics.)

110

Photon detectors The other category of infrared detectors is the quantum or photon detectors. In photon detectors, incident infrared photons result in the production of free charge carriers in the responsive element. No serious temperature change in the element takes place during this process. The above category can be further divided in four underlying processes: 1. Photoconductive effect. The principle behind this effect is that a change in the number of incident photons reaching a semiconducting material changes the number of the free charge carrier in the materials. Since electrical conductivity is directly proportional to the number of these charge carriers, it can be used to deduce the number of incident photons on the semiconductor. 2. Photovoltaic effect. In this case, a change in the number of incident photons on a semiconductor p-n junction results in a change in the voltage generated by the junction. Figures 6.5 and 6.6 show the energy band models for unilluminated and illuminated p-n junctions. 3. Photoelectromagnetic effect. In this case, the separation of the charge takes place via the use of a magnetic field. The charge separation produces a voltage that is directly proportional to the number of incident infrared photons. 4. Photoemissive effect. In this case, an incident photon is absorbed by the surface and gives up its energy to a free electron. This electron can escape the surface and in the case that the surface is in an evacuated chamber equipped with an anode and an eternal circuit, electric current is detected.

Band _.-----------Fermi Level

P-l

n-Region

Unilluminated p-n junction Fig. 6.5. Energy band model for an unilluminatedp-n junction. (Reproduced from Ref. [4] with permission. Copyright 1978, Office of Naval Research.) 111

Conduction Band

h

-'

h f

Illuminated p-n junction Fig. 6.6. Energy band model for an illuminated p-n junction. (Reproduced from Ref. [4] with permission. Copyright 1978, Office of Naval Research.)

6.3 DETECTOR NOISE

There are several contributors to the noise (the fluctuation in signal intensity for a steady radiation field) of an infrared detector. In infrared spectroscopy, the detector noise is most often much higher than any other noise source. In addition, it has usually a thermal origin. Therefore, the majority of infrared detectors do not operate at room temperature. Johnson noise: this type of noise is generated in resistors due to the random thermal motion of the charge carriers. As the temperature of the element increases there is a concurrent increase in the average kinetic energy of the carriers, which results in an increased electric noise voltage. This is the reason this type of noise is also called thermal noise. Shot or Schottky noise: this is random noise that has to do with the statistical fluctuations of the photon fluxes. It has its origin in the discreteness of electrical charge. Both of the above types of noise are 'white' types of noises. This means that they are independent of the frequency all the way out to the cut-off frequency. Other types of noise exist that are dependent on the frequency of the incoming radiation. The most important of these types is the 1/f noise. Its mechanism is not well understood but, as the name implies, its magnitude is reversibly proportional to the frequency of the radiation. 112

In addition, electronic components associated with the detector contribute to the noise. Pre-amplification is always required for any type of detection system.

6.4 PERFORMANCE OF AN INFRARED DETECTOR The performance of an infrared detector is measured by a set of figures of merit. The responsivity of the detector is the output of the electrical signal to the incident radiation power. The noise equivalent power (NEP) is the level of incident infrared signal that produces a signal-tonoise ratio of one. Detectivity is defined as the reciprocal of the noise equivalent power (NEP). The normalized detectivity D* is a widely used figure-of-merit and it includes the area of the detector and frequency bandwidth of the measurement. Thermal detectors show a 'flat' detectivity response throughout the entire spectral region. In contrast, photon detectors have higher detectivity but over a limited spectral range. Figure 6.7 shows the

10

lA

I I

E

in

8

1_1_~~~' N~I

I

1~~,___

i.. 2

4

6

8 10 12 Wavelength (rim)

14

16

18

9109 i

10 X

10 10

3 2 10 10 Chopping Frequency (cps)

10

Fig. 6.7. Plots of D* versus wavelength for (a) thermocouple versus (b) an InSb detector. (Reproduced from Ref. [4] with permission. Copyright 1978, Office of Naval Research.)

113

l

10'

1011

1010

10

9

1081

Wavelength (m)

Fig. 6.8. D*(X) values for a number of commercially available quantum detectors. (Reproduced from Ref. [4] with permission. Copyright 1978, Office of Naval Research.)

response for a thermocouple detector as compared with the spectral response for an InSb photon detector. In addition, Figure 6.8 shows the plots of the normalized detectivity D* for a collection of commercially available detectors. 6.4.1 MCT detector One of the most widely used infrared detectors is the mercury cadmium telluride (MCT) detector. This is a photon detector which needs to operate at liquid nitrogen temperatures of 77 K. Figure 6.9 shows the spectral response of the commercially available detectors as a function of wavelength. On the other hand, Fig. 6.10 depicts the frequency response of this detector's D*. Essentially, the response is 'flat' from about 103 Hz to 106 Hz. Furthermore, the alloy composition deter114

I

I

10 M

10

2

Z

1

M

9

0

- 10

4

,

t I 6

I

I'

I

1(6

8 10 12 14 Wavelength (m)

Fig. 6.9. Plot of detectivity versus wavelength for MCT detectors. (Reproduced from Ref. [4] with permission. Copyright 1978, Office of Naval Research.)

..

. _

1 10 2 10 RF

109 2 -

-8

In

. . ... . . .. '

I

........

2

103

..

.

I ~{J· I

10 4

1

..

I~~ I{

105

{ II~

106

107

Frequency (Hz)

Fig. 6.10. Plot of D* versus frequency for an MCT detector. (Reproduced from Ref. [4] with permission. Copyright 1978, Office of Naval Research.)

W W C 0..

i.

10

a:

IB\

.

C

i

t

O

C:

- Spectral Response

is a Function of x I

i 6

l 8

l 10

I 12

, 14

i 16

l 18

Wavelength (. m)

Fig. 6.11. MCT detector. Effect of alloy composition to the spectral responsivity characteristics. (Reproduced from Ref. [4] with permission. Copyright 1978, Office of Naval Research.)

115

x

Fig. 6.12. Wavelength cut-off for an MCT detector versus alloy composition at 77 K (Reproduced from Ref. [4] with permission. Copyright 1978, Office of Naval Research.).

mines the spectral response of the detector element. Figure 6.11 shows the spectral responsivity for three different alloy compositions (Hg,,CdxTe).

It is evident that manipulation of the alloy composition results in a detector which is tailored to particular needs for wavelength sensitivity. The spectral response of the material is determined by the energy gaps between the various energy levels in the material. Therefore, the energy gap in an MCT alloy is related to the ratio of HgTe to CdTe. Figure 6.12 shows the wavelength cut-off for this ternary alloy system.

6.5 OTHER COMPONENTS 6.5.1 Beamsplitters and mirrors The typical kind of beamsplitters in the mid-infrared region on commercially available instruments is of the germanium (Ge) on potassium bromide (KBr) substrate type. Recently, semiconductor film beam116

t5

D Wavelength (m)

Fig. 6.13. Reflectance of some common metallic films used as mirrors. (Reproduced from Ref. [12] with permission. Copyright 1957, Wissenschaftliche Verlagsgesellschaft mbh.)

splitters for infrared spectrometers have been described which are formed from self-supporting semiconductors, including carbon films. Preferably, the beamsplitters are formed from silicon, germanium, or diamond films [11]. In addition, Figure 6.13 shows the reflectance for some commonly used mirror surfaces in the infrared region [12].

6.6 DISPERSIVE INSTRUMENTS Nowadays, dispersive instruments are used only in selected applications due to the fact that interferometric instruments offer distinct advantages for most applications. However, there are places where dispersive instrumentation is still used when the response at one wavelength or a short range of wavelengths is sought [13]. The basic components of a dispersive spectrometer are the same as in a Fourier transform instrument with the exception of the interferometer. Any differences are within the elements and are the result of the different ways that the source radiation is detected. For instance, sensitive thermocouple detectors are commonplace in dispersive instruments, whereas they are not appropriate for rapid-scanning 117

instruments [14]. In a dispersive instrument a monochromator is used in the place of the interferometer. Before 1950, the monochromator was a rock salt prism for use in the mid-IR region of the electromagnetic spectrum and later was replaced by a diffraction grating. Older monographs provide excellent background information on the operation and maintenance of dispersive infrared instruments and the reader is recommended to consult them [15,16].

6.7 MICHELSON INTERFEROMETER Most commercial interferometers are based on the original Michelson design of 1891 [17]. Interferometers record intensity as a function of optical path difference and the produced interferogram is related to the frequency of the incoming radiation by a Fourier transformation. The principle of a Michelson interferometer is illustrated in Fig. 6.14. The device consists of two flat mirrors, one fixed and one free to move, and a beamsplitter. The radiation from the infrared source strikes the beamsplitter at 45°. The characteristic property of the beamsplitter is that it transmits and reflects equal parts of the radiation. One classic type of beamsplitter, useful in the mid infrared spectral region, consists of a thin layer of germanium (refractive index, n = 4.01) on an infrared transparent substrate (e.g., KBr). The transmitted and reflected beams strike the above described mirrors and are reflected back to the beamsplitter where, again, equal parts are transmitted and reflected. As a consequence, interference occurs at the beamsplitter where the

RADIATIO FROM SOt

MOVING MIRROR I M

BE

TO DETECTOR Fig. 6.14. Block diagram of a Michelson interferometer. 118

Un

Z l

M UI

U C I

I~

$---------

_

I

. .

Z'

U

'

0X o

Qc

I

"o00

SZS9

s=a

i2

-at 0

s TA POINTS

sea

s

5

7Z'-

75

Fig. 6.15. Typical interferogram showing centre-burst region.

radiation from the two mirrors combine. As shown in Fig. 6.14 when the two mirrors are equidistant from the beamsplitter constructive interference occurs for the beam going to the detector for all wavelengths. In this case, the path length of the two beams in the interferometer are equal and their path difference, called the retardation (6), is zero. The plot of detector response as a function of retardation produces a pattern of light intensity versus retardation, commonly referred to as the interferogram. The interferogram of a monochromatic source is a cosine function. Equation (6.1) describes the above relationship: 1(6) = B(v) cos(27v6)

(6.1)

where v is the wavenumber in cm-l and 6 is the optical path difference, or retardation. The Fourier transform of the above expression is a peak at the frequency of the monochromatic radiation. I is the intensity in the output beam as a function of retardation (6), and B is the intensity as a function of radiation frequency (v). In contrast, the interferogram of a polychromatic source can be considered as the sum of all cosine waves that are produced from monochromatic sources. The polychromatic interferogram has a strong maximum intensity at the zero 119

retardation point where all the cosine components are in phase, as can be seen in Fig. 6.15. This point is also known as the centreburst point [18]. The expression for the intensity of the interferogram of a polychromatic source as a function of retardation is described by Eq. (6.2): 1(6) = [[B(v)[1+ cos(2cv6)]] /2]dv

(6.2)

Thus, in Fourier transform interferometry the data are "encoded" by the interference produced by the retardation and then "decoded" by the Fourier transform to yield the desired intensity signal as a function of frequency (or wavelength).

6.8 ADVANTAGES OF INTERFEROMETRY Two kinds of multichannel advantages exist in Fourier transform interferometry, compared with a dispersive instrument in which only a very narrow band of frequencies is observed at a time. The first-and biggest practical advantage of Fourier transform spectroscopy-is the simultaneous detection of the whole spectrum at once; it is called the Fellgett or multiplex advantage [19,20]. Even though a factor of ca. 2 in signal strength is lost because half of the beam is reflected back to the source, the multichannel advantage is nevertheless 10 4 or higher. That is, theoretically an interferometer can achieve comparable signal-tonoise to a dispersive monochromator 104 faster. In addition, the so-called Jacquinot or 'etendue' advantage exists. This advantage is associated with the increase in source throughput [21]. During dispersive detection the throughput is severely limited by the area of the entrance slit. Even though the interferometer has an entrance aperture of its own, its throughput advantage ranges from 10 to 250 over the infrared frequency range. This was the reason that FTIR spectra of astronomical sources, where very weak astronomical emission sources are present, were produced even before the Fast Fourier Transform (FFT) was invented [22]. A third practical advantage of interferometry is the so-called Connes or registration advantage. Connes advantage stems from the ability of interferometry using a monochromatic source (e.g. a heliumneon (HeNe) laser in today's spectrometers) to accurately and precisely index the retardation, resulting in a superior determination of the 120

retardation sampling position. For example, if the above mentioned HeNe laser ( = 632.8 nm) is used, zero crossings in the visible interferogram occur at intervals of 632.8/2 nm = 0.3164 mm. Because the Nyquist theorem demands at least two sampling points per cycle, the highest infrared frequency that would satisfy the Nyquist criterion is 15,804 cm -l. For mid-IR use, sampling at every other zero-crossing (1 kHeNe intervals) produces a maximum Nyquist frequency of 7902 cm l. Connes advantage allows tremendous reproducibility of interferogram sampling and data storage. This results in full realization of signal-tonoise problems from repeated scans and it is particularly useful for the dynamic experiments that will be discussed later in this book [23,24].

6.9 APODIZATION The amplitude of the side lobes which appear adjacent to absorption bands in the Fourier transform of an interferogram can be drastically reduced if a mathematical manipulation is performed. This treatment is called apodization, from the Greek word ano6os (without feet). This mathematical treatment is necessary because the Fourier transformation is performed over finite limits, even though the theoretical expression for the interferogram's intensity involves infinite limits. Therefore, when the interferogram is truncated, this sudden cut-off results in the appearance of oscillations around the sharp spectral features (absorption bands) in the transform. When the interferogram is multiplied by the apodization function, the transform is essentially free of side lobes. Two of the most popular and effective apodization functions are the triangular and the HappGenzel functions. The amplitude of the first side lobe using triangular apodization is larger than that of the Happ-Genzel function, but the opposite is observed for the subsequent lobes. In general, it can be stated that Happ-Genzel apodization is quite similar to triangular apodization and for most situations they give comparable results [25]. Both types of apodization were used at different times in the work reported in this dissertation. Overall, it can be stated that the biggest drawback of apodization is the worsening of the spectral resolution, since the contributions of the extremes of the interferogram wings are reduced. Therefore, a trade off exists between the reduction in spectral distortion and the worsening of resolution. 121

6.10 RESOLUTION Resolution is defined as the minimum distinguishable spectral interval. The maximum retardation determines the resolution of the scan. The maximum optical resolution achievable by a particular FT-IR spectrometer is given by (Dma) - cm l, where D.ma is the maximum optical path difference attainable by the interferometer. Figure 6.16 shows the effect of different resolution on the appearance of single beam spectra. The spectra shown in this figure have been offset for clarity. Figure 6.16a shows the open-beam background spectrum of the unpurged spectrometer recorded with a resolution of 2 cm - l (Dm~ = 0.5 cm). It is clear that the rotational lines of vapour water are well resolved. In contrast, Fig. 6.16b shows the open-beam spectrum acquired with 16 cm - l resolution (Dm, = 0.0625 cm). Vapour water absorptions are not resolved due to the lower resolution. Dynamic methods have the potential to increase spectral resolution beyond the above limit due to the existence of the possibility of different responses of the components of highly overlapped bands. This possi-

a)

Ca

E M Mt U e

=

.

'A

b)

3600

'100

2600

2100

1600

1'0O

600

Wavenumbers Fig. 6.16. (a) Open-beam background, 2 cm-l; (b) open-beam background, 16 cm 1 .

122

bility will be further discussed when dynamic infrared experiments will be presented.

6.11 PHASE CORRECTION The non-ideality of the beamsplitter in a real interferometer results in the introduction of sine components to an interferogram which, in principle, should consist only of cosine components. Equation 6.3 shows the modified relation for the intensity of the interferogram: +o

1(5) =

[[B(v)[l

+ cos(2gv6 + DBS(v))]] / 2]dv

(6.3)

where FBs(V) is the wavelength dependent phase shift introduced by the beamsplitter. Phase correction is the mathematical procedure to remove the sine components from the interferogram. The Fourier transform of a complete double-sided interferogram provides the correct power spectrum, without any phase correction, since the ambiguity does not affect the magnitude. However, when a single-sided interferogram is computed, some knowledge of the phase is required in order to compute the true spectrum [26]. Two of the most popular phase correction routines used in single-sided interferograms are the Mertz algorithm and the Forman algorithm. In the Mertz routine, the largest data point in the interferogram is assigned as the zero retardation point and the amplitude spectrum is calculated with respect to this point. A short double-sided interferogram is measured and its corresponding phase array is used to phase correct the entire single-sided spectrum. The Forman correction is essentially equivalent to the Mertz routine but it is performed in the retardation space [27,28]. Modifications to the Mertz phase correction have appeared in the literature and were originally applied to the vibrational circular dichroism (VCD) spectra [29]. The result of these modifications is that the phase spectrum does not change sign if a quadrant boundary is crossed. As an alternative, a "stored" phase array can be used to produce proper phase correction for the transformed interferograms. This phase array is calculated from a double-sided reference interferogram. The procedure relies on the fact that the beamsplitter phase does not change from scan to scan. 123

6.13 EFFECTS OF MIRROR MISALIGNMENT Mirror misalignment in an interferometer produces a lowering in the energy on the higher energy end of the spectrum (transform). This can be seen in Fig. 6.17 which illustrates three single beam intensities. The degree of alignment varies in these three scans, and this is evident on the high energy portion of the spectrum. These continuous-scan results can be compared to step-scan phase modulation results obtained by dithering the moving mirror with piezoelectric transducers. Continuous-scanFT-IR Most commercially available FT-IR spectrometers use the continuousscan mode of operation, where the moving mirror is scanning at constant velocity. This type of scanning works very well for routine measurements. In the continuous-scan mode of interferometry the laser fringe counter is used to sense the accuracy of the scanning velocity. If a deviation is sensed, correction signals are generated that

Wavenumbers Fig. 6.17. Effect of mirror misalignment on the appearance of single-beam transmission spectra. 124

assure the proper operation (constant velocity). The consequence of this mode of operation is that each infrared wavelength (), is modulated at its own particular Fourier frequency, given by Eq. (6.4): f(k) = 2v/

(6.4)

where v is the mirror velocity. Continuous-scan FT-IR is the technique of choice when static spectral properties are determined. Co-addition of successive scans increases the signal-to-noise ratio (S/N) by a factor proportional to t, where t is the time that the signal is averaged at each collection point. Step-Scan FT-IR In step-scan FT-IR data are collected while the retardation is held constant or is oscillated about a fixed value. Therefore, in order to apply the technique to mid-infrared and shorter wavelength measurements, a method for controlling the retardation and implementing a special sampling rate comparable to that achieved in modern continuous-scan instruments is required. In recent years several different control methods have been reported. However, all of these basically rely on the use of the HeNe laser fringe pattern to generate the control signal and to determine the step size [30]. The biggest advantage of the step-scan mode is the separation of the time of the experiment from the time of the data collection. Two types of experiment are possible with step-scan interferometry. One type is the time-domain or time resolved experiments where data are collected as a function of time at each mirror position. Sorting of the data results in interferograms that contain spectral responses at different times. The event under study must be a repeatable process in order for the experiment to work. The other type of experiment capable with step-scan is the so-called frequency domain or synchronous modulation experiments. In these experiments, there are two ways to modulate the intensity of the infrared radiation in order to generate step-scan interferograms. One way is to use amplitude modulation (AM) which can be achieved by means of a chopper. When a chopper is used for intensity modulation, a lock-in amplifier is used to detect the signal before digitization occurs. The technique has the drawback that the signal is riding on top of a large DC offset, which has to be subtracted before any meaningful data can be obtained. This can be done by either calculating the average 125

value of the interferogram and subtracting it from each sample point before the Fourier transform takes place or by setting the lock-in amplifier offset to zero, far from the interferogram. Even though the latter technique eliminates the problem of reduced dynamic range, the technique is still susceptible to DC drift. Another way to achieve modulation of the radiation is by phase modulation (PM). Phase modulation is achieved in some step-scan instruments by a low amplitude oscillation of the moving mirror along the light path, but any other way of producing phase-difference modulation is acceptable. PM results are superior to AM results by at least a factor of 2 in S/N, when the experiment is detector-noise limited. This improvement stems from the fact that the PM interferogram is essentially the first derivative of the AM interferogram, therefore source intensity fluctuations and other variations of the beam intensity will cancel out [31]. Another parameter associated with PM modulation experiments is the so-called 'phase modulation characteristic' [32]. This refers to the connection between the amplitude of the phase modulation and the wavelength region of maximum modulation efficiency. For the mid-IR region, a PM modulation amplitude of 2 XHeNe (zero-to-peak) is appropriate (maximum modulation at 2300 cm-l). In contrast to the continuous-scan method, the advantages of stepscan operation include the ability, as mentioned above, to apply virtually any modulation frequency to the infrared radiation and to carry out multiple modulation experiments. Since the frequency of modulation is not a function of any retardation velocity (e.g., mirror scan speed), they have no dependence on radiation wavelength. In addition, the use of lock-in amplifier detection or digital signal processors (DSP), provides a high degree of noise rejection, analogous to the Fourier filtering effective in the continuous scan mode. Another advantage of lock-in amplifier or DSP detection is the easy retrieval of the signal phase. This is possible due to the fact that the beamsplitter (instrumental) phase is identical for the in-phase and quadrature (900 out of phase) components of the signal. These components are easily obtained as outputs of a two-phase lock-in amplifier. As a result, not only the magnitude M, but also the phase D can be easily obtained by following Eqs. (6.5) to (6.8): M = (2 + Q2 )1/ 2

(6.5)

D = arctan(Q/I)

(6.6)

126

I=Mcos¢

(6.7)

Q = M sine

(6.8)

The signal-to-noise (S/N) ratio is increased by staying longer at each data collection point. The two modes of operation, step-scan and continuous-scan, should produce identical results under conditions of detector-limited noise. The time resolution of the step-scan technique is limited only by the rise-time of the detector, by the electronics (especially by the A/D converter), and by the signal strength. Therefore, it is capable of measuring various relaxation processes that occur in the sub-microsecond regime and are closely associated with molecularscale phenomena.

6.14 FOURIER TRANSFORMATION AND ITS USE IN FT-IR INSTRUMENTATION The breakthrough in the application of interferometry to spectroscopy came with the discovery by Cooley and Tukey of the fast Fourier transform (FFT) algorithm in 1964 [33]. The FFT procedure takes advantage of several properties of the discrete FT, which is somehow redundant in nature [34]. The FT case can be represented as an nvector (n points interferogram) which must be multiplied by an nxn matrix, each row of which is a discrete representation of a complex sinusoid. The result of the multiplication is an n-vector, which is the transformed spectrum. A straightforward approach requires n2 operations, where operations are complex multiplications followed by complex additions. Since the nxn matrix is highly ordered and cyclical it can be readily factored. IfNis chosen such that it is an integral power of two, then extra advantage may be realized by calculating the FT on a digital computer [35]. As an example, the Fourier transform of a 2048 point vector requires (2048)2, or 4.2 million multiplications. The FFT algorithm reduces this amount to (2048) x log(2048), for a total of 24233 multiplications. Obviously, the great advantage of FFT can be appreciated as the number of data points gets larger and larger. Today's personal computers can calculate an array similar to the one described above in a fraction of one second.

127

REFERENCES 1. W.F. Herschel, Phil. Transact. Roy. Soc., 90 (1800) 284. 2. C.N.R. Rao, Chemical Applications of Infrared Spectroscopy. Academic Press, New York, 1963. 3. J.F. Ferraro, Spectroscopy, 14(2), 1999. 4. W.L. Wolfe and G.J. Zissis, The Infrared Handbook. Office of Naval Research, Department of the Navy, Washington, DC, 1978. 5. J.C. Morris, Comments on the measurement of the emmitance of the globar radiation source. J. Optical Soc. Am. (Washington, DC), 51 (1961) 758. 6. Nicolet Instruments, private communication. 7. W.Y. Ramsey and J.C, Alishouse, A comparison of infrared sources. Infrared Physics (Pergamon, Elmsford, NY), 8 (1968) 143. 8. W.L. Wolfe and G.J. Zissis, The Infrared Handbook. Office of Naval Research, Department of the Navy, Washington, DC, 1978, Chapter 11. 9. M.J.E. Golay, Rev. Scient. Instrum., 18 (1949) 347. 10. M. Hercher, Detectors and Lasers, in: Contemporary OpticalEngineering. The Institute of Optics, Rochester, NY. 1976 11. A. Simon, J. Gast (Bruker Analytische Messtechnik GmbH, Germany. Patent written in German. Application: DE 92-4242440 921216 12. G. Hass and A.F. Turner, in: M. Auwarter (Ed.), Coatings for Infrared Optics. Wissenschaftliche Verlagsgesellschaft mbh, Stuttgart, 1957. 13. I. Noda, A.E. Dowrey and C. Marcott, Appl. Spectrosc., 42 (1988) 203. 14. R. White, Chromatographyl/FourierTransformInfraredSpectroscopy and its Applications. Marcel Dekker, New York, 1990. 15. C.N.R. Rao, Chemical Applications of Infrared Spectroscopy. Academic Press, New York, 1963. 16. A.L. Smith, Applied Infrared Spectroscopy, Fundamentals, Techniques, andAnalytical Problem-Solving. Wiley-Interscience, New York, 1979. 17. A.A. Michelson, Phil. Mag., Ser. 5, 31 (1891) 256. 18. P.R. Griffiths and J.A. de Haseth, Fourier Transform Infrared Spectroscopy. Wiley, New York, NY, pp. 407-425, 1986. 19. P. Fellgett, Thesis, Cambridge University, Cambridge, UK, 1951. 20. P. Fellgett, J. Phys. (Radium), 19 (1958) 187. 21. P. Jacquinot, J. Opt. Soc. Am., 44 (1954) 761. 22. J. Connes and P. Connes, J. Opt. Soc. Am., 56 (1966) 896. 23. J. Connes, Optical Instruments and Techniques. Oriel Press, Paris, 1970. 24. R.A. Palmer, Spectroscopy, 8(2) (1993) 26. 25. A.G. Marshall and F.R. Verdun, FourierTransforms in NMR, Optical, and Mass Spectrometry. Elsevier, Amsterdam, 1990. 26. L. Mertz, InfraredPhys., 7, (1967) 17. 27. M.L. Forman, J. Opt. Soc. Am., 56 (1966) 978. 128

28. 29. 30. 31. 32. 33. 34. 35.

M.L. Forman, W.H. Steel and G.A. Vanasse, J. Opt. Soc. Am., 56 (1966) 59. C.A. McCoy and J.A. de Haseth, Appl. Spectrosc., 42 (1988) 336. R.A. Palmer, Spectroscopy, 8(2) (1993) 26. J. Chamberlain, InfraredPhys., 11 (1971) 25. J. Chamberlain and H.A. Gebbie, Infrared Phys., 11 (1971) 57. J.W. Cooley and J.W. Tukey, Math. Comput., 19 (1965) 297. C.J. Manning, Ph.D. Thesis, Duke University, 1991. M.L. Forman, J. Opt. Soc. Am., 56 (1966) 978.

129

Chapter 7

Sampling techniques and applications

PART A. DIRECT TECHNIQUES In recent years there have been remarkable theoretical and experimental advances in a variety of measuring methods for infrared spectroscopy, which have made it possible for anyone to try various infrared measurement methods relatively easily [1-6]. While transmission methods are most often used for measurements of infrared spectra of general samples, the attenuated total reflection (ATR) method, diffuse reflectance (DR) method, reflection-absorption (RA) method, photoacoustic spectroscopy (PAS), infrared microspectroscopy, and emission spectroscopy, etc., are also often employed.

7.1 TRANSMISSION SPECTROSCOPY Transmission methods are suitable for infrared measurements of liquid (solutions) samples, powder samples and gases. Practical measures for the intensities of infrared bands in a transmission spectrum are given by Lambert-Beer's law. Before we explain the law, let us define transmittance, T, and absorptivity, A, When a parallel light beam enters transparent material with the length of d cm (as shown in Fig. 7.1), transmittance, T, is defined as follows: T=ItIIo

(7.1)

where It and Io are the intensities of the parallel beam at the positions of incidence and exit, respectively. Transmittance, T, is the proportion of the intensity of transmitted light to that of incident light. On the 131

! Io

x

-

x+dx /::

It

>

Fig. 7.1. Absorption of light by a transparent material.

other hand, absorptivity, A is the proportion of the intensity of absorbed light (a) to that of incident light; namely A = I/Io

(7.2)

If one assumes that the transparent material does not scatter light and is non-fluorescent, the following equation holds: It+a

=Io

(7.3)

Therefore A+T=I

(7.4)

Now we will develop Lambert-Beer's law which explains the degree of decrease in the intensity of incident light. If the intensity of incident light decreases from i to i + di when the light proceeds from x to x + dx in the transparent material, the degree of decrease in the intensity (di) is considered to be proportional to the intensity of the incident light (i), the concentration of a sample, c (mol dm-3 ), and the thickness of the sample (dx): -di = c*ic dx

(7.5)

where e* is a proportional constant. If we make definite integral of Eq. (7.5) from 0 to d 132

-

i =d c|cf dx

(7.6)

I t = Io exp(-cdc*)

(7.7)

Thus

Equation (7.7) is a mathematical representation of Lambert-Beer's law, which means that the intensity of incident light decreases exponentially. Lambert-Beer's law is usually used in the form of a common logarithm. Equation (7.7) can be expressed as follows in terms of a common logarithm: It = Io.lO0 d

(7.8)

where 8

= 0.434£*

(7.9)

If we take the common logarithm of Eq. (7.8) log(IJIt) = log(l/T) = cds

(7.10)

log(IJI t) is defined as absorbance, which is proportional to the concentration, c, of the sample and the path length, d, of a cell. The proportional constant is a measure of the strength of absorption at a particular frequency and is called molar absorption coefficient. The unit of 8 is (mol dm- cm) - l, namely, M-l cm-l (in SI units, it is m2 mol-). c is specific for each sample (the values of £ usually range from 10 to 105). One can calculate the concentration from the measurement of the absorbance of a sample, if one knows e and d (usually, d is taken as 1 cm). In contrast, one can determine from known d and c. When we use a transmission method on a liquid sample, the selection of window materials (which transmit infrared light) and the length of a cell are important. During selection of window materials, we must note a usable wavenumber range, a refractive index, and solubility in water. Table 7.1 summarizes the usable wavenumber range, the refractive index, and the solubility in water of representative window materials. Among the window materials for non-aqueous solutions, KBr is most often used. KBr has a wide usable wavenumber range 133

TABLE 7.1 The usable wavenumber range, the refractive index, and the solubility in water of representative window materials for infrared transmission spectroscopy Window material

Usable wavenumber range/cm - 1

Refractive index

Properties

CaF 2

75000-1000

1,40

insoluble in water

BaF 2

67000-800

1,45

insoluble in water

KBr

43000-400

1,52

soluble in water and alcohol

CsBr

42000-250

1,65

soluble in water

CsI

42000-200

1,72

soluble in water

NaCl

40000-600

1,50

soluble in water

KC1

33000-400

1,47

soluble in water

ZnSe

20000-500

2,42

insoluble in water

KRS-5

12000-350

2,35

insoluble in water, poisonous

Si

10000-100

3,42

insoluble in water

Ge

5000-400

4,00

insoluble in water

(4000-340 cm-l) and is inexpensive; furthermore, the refractive index of KBr (n = 1.52) is close to that of a solution. Although we can use only above 1000 cm -1, insoluble CaF 2 (n = 1.39) is proper for an aqueous solution (n = 1.33). KRS-5 (n = 2.37) is often used for infrared measurements in lower wavenumber regions. However, KRS-5 is poisonous and requires an attention. 7.1.1 Liquid (solution) samples Infrared spectra of liquid samples are measured using a fixed cell for liquid or an assembled cell. Fixed cells and assembled cells are basically the same in that a spacer having a certain thickness is disposed between two window materials. While a fixed cell is advantageous in that the cell has a clearly defined thickness and is airtight, it is not easy to clean this type of cell. Conversely, although an assembled cell is easy to clean, this type of cell is not airtight in general. When our sample is a solution sample with high viscosity and a low vapour pressure, it is also possible to measure a spectrum only with the solution sample held between two window materials. 134

Since water exhibits strong infrared bands, a path length of a cell must be 10 upm or shorter to measure an infrared spectrum of an aqueous solution by a transmission method. If we use heavy water (D 2 0) instead of light water (H 2 0), strong water bands (-3500 cm -l , -1650 cm- ) show a downward shift. Therefore, it is effective to use deuterium water; one measures spectral regions in the vicinity of 3500 cm -1 and 1650 cm-l. The ATR method is often also used to measure infrared spectra of aqueous solutions. A variety of liquid transmission cells are commercially available. Figures 7.2a and b show expanded views of the micro demountable flow-thru cell and heated demountable cell kit, respectively.

(a)

(hi

Fig. 7.2. Expanded views of the micro demountable flow-thru cell (a) and heated demountable cell kit (b). (From catalogue of Nicolet.) 135

7.1.2 Powder samples To measure an infrared spectrum of a powder sample by a transmission method, the KBr disc method is usually used. Since a particle diameter is too large when we use a powder sample as it is, incident light is irregularly reflected and, as a result, we cannot obtain a good spectrum. Therefore, it is necessary to crush the sample into pieces each having a diameter of 1-2 pm in advance. 7.1.3 KBr method When using the KBr method, crush and mix 1-2 mg of a sample and approximately 100 mg of KBr powder in an agate mortar, introduce the mixture into a tabletting equipment, and tablet. With this method, the following three points should be noted: 1. Since it is impossible to obtain good tablets if KBr or a sample contains moisture (i.e., resultant tablets become opaque), remove the moisture as much as possible. 2. If the sample is not sufficiently crushed, the tablets produced will become opaque or the quantity of transmitted light will decrease (due to light scattering). Hence, the crushing must be sufficient. 3. K+ or Br and cations or anions contained in the sample sometimes exchange with each other. Further, crushing, pressurization, etc. sometimes causes denaturation of the sample. To confirm that a situation like (3) has not occurred, it is desirable to measure infrared spectra of the same sample by other methods (e.g., the nujor method) in parallel with the KBr method. When the amount of the sample is small, use micro-tabletting which allows one to make a tablet having a diameter of 1 mm.

7.2 INTERNAL AND EXTERNAL REFLECTION SPECTROSCOPY 7.2.1 Attenuated total reflection (ATR) method As its name suggests, the ATR method is a type of reflectance method and has been used mainly for surface analysis and analysis of bulk materials and aqueous solutions [1-6]. In the ATR method, a sample is placed on a prism which has a larger refractive index than that of the sample, as shown in Fig. 7.3, and infrared light is introduced to the 136

epithelium

nucleus

anterior capsule

ncex - posterior capsule

cortex

ATR prism (ZnSe)

Fig. 7.3. An ATR prism and an eye lens on it. (Reproduced from Ref. [7] with permission. Copyright (1998) Society for Applied Spectroscopy.)

prism such that the infrared light is reflected totally at an interface between the prism and the sample [7]. At this stage, to develop total reflection, the incident angle 0 must be larger than the critical angle Oc. Where the refractive indices of the prism and the sample are nI and n2, respectively, the critical angle is defined as: Oc = sin-l(n2 /nl)

(7.11)

Total reflection discussed here is not reflection whereby the incident light is totally reflected at the interface, but reflection whereby the incident light penetrates inside the sample once to a certain extent before being reflected. That is, although the incident light is reflected 100% in a wavelength range in which the sample does not absorb light, in a wavelength range in which the sample absorbs it the reflectance is decreased depending on the strength of the absorption. Hence, if we measure the intensity of reflected light in a certain wavelength range, we can obtain a spectrum that resembles a transmission spectrum. In the ATR method, the penetrationdepth c is important. The depth dp is expressed as follows: dK

(7.12) 2x

in 2 0 _

)2)

where kl denotes the wavelength of the light within an ATR crystal. As is clear from the Eq. (7.12), the depth dp is determined by the incident angle, the wavelength, and further, by a ratio of the refractive index of 137

a sample to the refractive index of the ATR crystal. Since the depth dp is at most a few gm in the infrared region, an ATR spectrum provides information regarding a surface and around the same. In the ATR method, the contact between a sample and an ATR prism must be very smooth. Figure 7.3 shows an example of non-destructive analysis of an eye lens by the ATR method [7]. The ATR method is very suitable to obtain infrared spectra of aqueous solutions. Figures 7.4a and b show infrared spectra of water measured with the transmittance method and ATR method, respectively [8]. A strong feature which appears in the range of 3600 to 3200 cm-l is due to the OH stretching vibrations, while a band in the vicinity of 1640 cm-l is assigned to the HOH bending vibration. A major problem in studying aqueous solutions comes from these two bands. Although the spectrum (Fig. 7.4a) is obtained using a very thin liquid cell (whose thickness is 10 lpm), we can find a strong band with absorbance of 1.5 or higher. The ATR method considerably weakens the band due to the OH stretching vibrations. This is because the shorter the wavelength, the smaller the penetration depth dp (Eq. (7.12)). In this manner, with the ATR method, we can suppress the intensity of the strongest band of water, and therefore, data processing for subtracting the infrared spectrum of water is relatively easy. Figure 7.5a shows an ATR infrared spectrum of oxidized cytochrome c (pH 9.3, 10wt%) in a phosphate buffer. In Fig. 7.5b, a

0 cd

0

E:

4000 3600 3200 2800 2400 2000 1600 1200 800

Wavenumber/cm -' Fig. 7.4. Infrared spectra of water measured with the transmittance method (a) and ATR method (b). (Reproduced from Ref. [8] with permission. Copyright (1994) Tokyo Kagaku Dojin.) 138

Z C3 C

EX

Wavenumber/cm -' Fig. 7.5. (a) An ATR/infrared spectrum of oxidized cytochrome c (pH 9.3, lOwt%) in a phosphate buffer. (b) An ATR/infrared spectrum of the phosphate buffer solution. (c) Difference spectrum obttained by subtracting spectrum (b) from spectra (a).

spectrum of the phosphate buffer solution is presented. Note that the spectrum of the protein solution is close to that. Hence, in order to extract useful information from the spectrum shown in Fig. 7.5a, we have to subtract the spectrum of buffer (Fig. 7.5b). Figure 7.5c shows a resulting spectrum that is obtained by subtracting the spectrum of buffer. A simple guideline for determining if the water spectrum is successfully subtracted is to see whether the range of 2400 to 1700 cm - ' is flat. 7.2.2 Prisms and accessories for ATR spectroscopy ATR spectroscopy has long been used for various samples from aqueous solutions to bulk materials [1-6]. A variety of prisms and accessories for ATR spectroscopy have been developed and are commercially available. They are divided into two groups: single reflection type with a hemicylinder crystal and multiple-reflection type with a trapezoidal crystal. Figure 7.6 depicts optical schematics of several commercially available ATR accessories. In FT-IR spectroscopy, the multiple-reflec139

tion setups are more popular than single reflection setups because in the former one can control the ATR signal intensity easily by adhering samples on both sides of an ATR prism or by changing the size of samples. The energy loss by the multiple-reflection can be compensated by the increase in the number of acquisition or the use of a MCT

(b)

(c)

Fig. 7.6. Various accessories for ATR spectroscopy: (a) single-reflection variable-angle hemicylinder; (b) multiple-reflection single-pass crystal; (c) circle ATR cell for liquid samples. TABLE 7.2 The usable wavenumber range, the refractive index, and the properties of representative window materials for ATR spectroscopy Window material

Usable wavenumber I range/cm

Refractive index

Properties

ZnSe KRS-5 AS 2 Se3

20000-400 20000-300 12500-800

2,4 2,40 2,80

Si Ge

5000-1600 5000-900

3,40 4,00

suitable for aqueous solution poisonous suitable for aqueous solution, fairly fragile fragile high refractive index, suitable for aqueous solution, fragile --

140

(a)

(b) i I i II I

I

Fig. 7.7. (a) Accessory for micro ATR; (b) in situ ATR accessory.

detector. The ATR cell shown in Fig. 7.6b is very suitable for films, solids, and liquids, while the cell shown in Fig. 7.6c is designed for liquid samples. Table 7.2 summarizes properties of selected optical materials used for ATR spectroscopy. Recently, micro ATR cells and in situ ATR accessories have made marked progress; Figs. 7.7a and b show their examples, respectively. 7.2.3 Examples of ATR infrared studies Two examples of ATR infrared studies are introduced here. Another example will be described in Section 9.7.3. Figure 7.8a-c show ATR spectra of water and the anterior and posterior portions of the lens capsule of a 3-month-old albino rabbit lens on an ATR prism [7]. The spectra of the capsules are very close to that of water, but one can see weak features in the 1600-1000 cm-l region that are not assigned to water. Figures 7.9a and b present difference spectra that were obtained by subtracting the spectrum of water from the spectra of the lens capsule [7]. The thickness of the capsules of the rabbit lens is about 2-20 pm, while the penetration depth of the ATR prism in the 1700-1000 cm-l region is less than 1 pm. Therefore, all the ATR signals 141

0031

w 0 z

0 o (,

WAVENUMBER /cm-

1

Fig. 7.8. ATR/infrared spectra of water (a) and the anterior (b) and posterior (c) portions of the lens capsule of a 3-month-old albino rabbit. (Reproduced from Ref. [7] with permission. Copyright (1989) Society for Applied Spectroscopy.)

in Fig. 7.9a and b may be due to type IV collagen, a major component of the capsule that envelops the lens. Figure 7.9c shows an ATR spectrum of purified type IV collagen in an aqueous solution. It is noted that the ATR spectra of the rabbit lens (Figs. 7.9a and b) are very close to that of type IV collagen in the aqueous solution (Fig. 7.9c). Bands at 1638 and 1553 cm-l are due to amide I and amide II of type IV collagen, respectively. Medium features at 1080 and 1033 cm -l are assigned to C-O stretching modes of collagen [7]. The frequencies of amide I and II suggest that type IV collagen in the capsules assumes a secondary structure that is very close to 3 helix [7]. This ATR study is a good example demonstrating the potential of ATR spectroscopy in investigating the structure of biological molecules in situ. ATR/IR spectroscopy has considerable promise in on-line monitoring [9,10]. In general, an ATR immersion probe is used as a sending head, which allows one to monitor reaction occurring within about 1 ipm of the surface of an ATR prism. The ATR method has two advantages in chemical process analysis. One is that the immersion probes have little effect from bubbles and suspensions in reaction mixture. Another is that the ATR method is applicable to samples that do not transmit enough light. Figure 7.10a shows the results of on-line monitoring of 142

WAVENMMBR /cm-1

Fig. 7.9. (a) The difference spectrum between spectrum (a) and spectrum (b) in Fig. 7.8. (b) The difference spectrum between spectrum (a) and spectrum (c) in Fig. 7.8. (c) ATR/ infrared spectrum of Type IV collagen from human placenta in aqueous solution (Reproduced from Ref. [7] with permission. Copyright (1989) Society for Applied Spectroscopy.)

the esterification of acid anhydride and diol measured by an ATR in situ setup [10]. The esterification is one of the most basic reactions in chemical industries. In the esterification by measuring the acid value and hydroxyl group value at any time the terminal point of the reaction must be predicted. It can be seen from Fig. 7.10a that as the reaction proceeds, a broad feature near 3400 cm l due to the OH stretching mode of alcohol decreases and a sharp band at 1700 cm-l assigned to the C=O stretching mode of ester increases. Figure 7.10b illustrates correlation for the acid value between the laboratory data and the ATR/ IR data [10]. 7.2.4 Reflection-absorption spectroscopy Reflection-absorption spectroscopy (RAS) is a powerful technique for measuring IR spectra of thin films on metal [1,2,8,11,12]. Figure 7.11 illustrates the principle of RAS. Let us consider the behaviour of polarized light impinging on a metal surface. When light impinges on 143

(a) initiation of reaction

0.5

_

N.2800

d -0.010 4000 3600 3200 !

41

-r

2400

fA"

1

200oo

160

1200

725

Q

a

0.5

C

_

in the middle of reaction -0.010

46)00 3600 3200

0.5

--(0

fi

----

2800 2400 2000

,iXjl 16

1200

725

termination of reaction

0003600

200

O WO

320

2800

200 2000

16t0

280

240

160

200

_, 12-0

72

Wavenumber cm ' (b) 120 Regression Statistics o00 RSO.=0.99 80 Std. Err.=3.4 40 - 20

0

6167.0

~~40l ~~~~3729 cJ

P~~~~~26 20 40 60 80 100 acid value iAoraor4 data)

36.1 26.6

25.9 120

Fig. 7.10. (a) The results of on-line monitoring of the esterification of acid anhydride and diol measured by an ATR in situ setup. (b) Correlation for the acid value between the laboratory data and infrared data. (Reproduced from Ref. [10] with permission. Copyright (1995) Kogyo Gijutsu Co.)

the metal surface, an electric field is generated near the surface. The intensity of the electric field depends upon the incident angle and the direction of polarized light. When perpendicular-polarized light is used, the electric vector of the incident light and that of the reflection light cancel each other out because the phase of the incident wave is shifted by 180 ° due to the effect of free electrons in metal. As a result, there is 144

Fig. 7.11. Behaviour of two kinds of polarized light which is incident onto a metal surface: (a) perpendicular polarization; (b) parallel polarization.

almost no standing wave. In contrast, when parallel-polarized light is applied, the electric vector of the incident light and that of the reflection light strengthen each other, generating an electric field perpendicular to the metal surface. The strength of this standing wave increases as the incident angle approaches 90 ° . Figure 7.12 shows the dependence of the absorption factors for the parallel (Ap) and perpendicular (As) polarization at the wavenumber of maximum adsorbed layer [12]. An optimal incident angle that gives the highest sensitivity changes with metals and the wavenumber of incident light. Usually the optimal incident angle is 85-89 ° . The higher the reflectivity of the metal, the larger the optimal incident angle. When parallel-polarized light impinges on a thin film absorbed on a metal surface, the standing wave interacts with the molecules in the film and the light is absorbed. The change in reflectance in the spectral region where the light is absorbed can be represented by the following equation [8]: -1 )

1 cs(7.13)

n

Cos0

Here, AR and Ro are a change in reflectance induced by the presence of the thin film and reflectance in the absence of the film, respectively. n1 and n2 are refractive indices of medium (such as atmosphere) and that 145

x

0

20

40

60

80

Angle of incidence, degrees Fig. 7.12. Dependence of the absorption factors for the parallel (Ap) and perpendicular (As ) polarization at the wavenumber of maximum adsorbed layer. (Reproduced from Ref. [12] with permission. Copyright (1993) John Wiley &Sons.)

thin film, respectively. 0, a, and d represent an incident angle, an absorption coefficient, and a thickness of the thin film, respectively. In the case of transmission spectroscopy, the ratio of a change in the intensity of transmitted light in the presence of the sample (A) and the intensity of the incident light (Io) can be represented by the following equation: rlIo = -ad

(7.14)

Comparison of Eqs. (7.13) and (7.14) reveals that the sensitivity of RAS is higher by (4n1 3 sin20)/(n2 3 cosO) than that of transmission spectroscopy [8]. The term 4nl3 sin 2 0 represents the dependence of the intensity of standing wave appearing on the metal surface upon the incident angle, while the term 1/cosO is concerned with a sample area irradiated by the incident light. RAS has been widely used for studies of thin films such as Langmuir-Blodgett films, thin polymer films, coating films and epitaxial layers on silicon wafers [12]. RAS is very useful not only for investigating chemical composition and molecular structures but also 146

for molecular orientations. Good examples of applications of RAS are given in Section 8.6.

7.3 DIFFUSE REFLECTANCE SPECTROSCOPY A diffuse reflectance method is a measurement method for measuring reflected light (diffuse-reflected light) which exits a sample surface after impinging upon the sample surface and thereafter getting reflected, transmitted while refracted, and scattered repeatedly (Fig. 7.13) [1,2]. When we irradiate a powder sample with light, while a portion of the light is regularly reflected at a powder surface, the remaining majority enters the powder and diffuses. During the process of diffusion, since light having particular wavelengths is absorbed by the sample, if we measure the intensity of the diffuse-reflected light at various wavelengths, we obtain a spectrum that is similar to a transmission spectrum. The intensity of a diffuse-reflectance spectrum is generally expressed by a Kubelka-Munk equation such as: K

(1-R

S

2R

)2

f(R

)

(7.15)

where K, S, and R, are an absorption coefficient, a scattering coefficient and an absolute diffuse reflectance, respectively. Further, f(R.) is called an K-M (Kubelka-Munk) function. Since K is in proportion to a I

I

I:D;incident light diffuse reflection light S: regular refrection light Fig. 7.13. Schematic representation of regular reflection and diffuse reflectance for a powder sample.

147

YA

'1 F

incident light

t

o

d

dy

diffuse reflectance light

1I

t J+dJ

I+dI

t/

t

0 Fig. 7.14. Incident light and diffuse-reflected light in a powder layer (Kubelka-Munk theory).

molar absorption coefficient, , and a concentration of a sample c (K = yc where y is a proportional constant), if the scattering coefficient is a constant value, the K-M function should be in proportion to the sample concentration. To measure the absolute diffuse reflectance is not realistic; we therefore measure a relative diffuse reflectance R instead (which is a ratio of the intensity of reflection from the sample to the intensity of reflection from a standard substance). Let us yield a Kubelka-Munk equation. Assuming that monochromatic light is incident upon a powder layer which has a thickness d, as that shown in Fig. 7.14, from the top along they-axis, the incident light propagates in the sample in the negative direction along the y-axis and diffuse-reflected light propagates in the positive direction along the same axis. A decrease, dI, in the intensity I of the light which propagates within a thin layer portion dy in the negative direction and a decrease, dJ, in the intensity J of the light which propagates in the positive direction are expressed as follows:

-dI = (K+ S) Idy - SJdy

(7.16a)

dJ = - (K+ S) Jdy + SIdy

(7.16b)

The first term on the right-hand side of Eq. (7.16a) denotes an intensity decrease due to absorption and scattering of the incident light, while the second term on the same side denotes a contribution from back 148

scattering light. Equation (7.16b) can be interpreted in a similar manner. Now, if(S + K)/S = 1 + K/S = a, Eqs. (7.16a) and (7.16b) can be re-written as follows: -dl/Sdy = aI- J

(7.17a)

dJSdy = -aJ+I

(7.17b)

Diving Eq. (7.17a) by I and Eq. (7.17b) by J and adding the two equations, we obtain dr/Sdy = r2 - 2ar+ 1

(7.18)

Integrating this equation, we obtain

(r2 -2ar +1)-ldr =Sldy

(7.19)

In this equation, the values of r when y = 0 and y = d hold are Rg (a background reflectance) and R (the reflectance of the sample), respectively. If an integral of Eq. (7.19) is expressed by the relationship as below dr 2

(r - 2ar + 1)

dr

~dr 1

r + (2ar - 1)2 }{r -(2ar - 1) 2})

(7.20)

the integral is solved simply as: {R, -a -(a 2 1)V2 {Rg -a +(a 2 _ 1)21 =2sd(a2 1)2 In12 2sd(a _ 1)"" {Rg -a -(a 2 -1)2 ){Rs -a +(a 2 -1)1/ 2

(7.21)

(7.21)

Since we can ignore Rg for an extremely thick sample (i.e., Rg = 0 when d = ~), Eq. (7.21) is simplified as follows: {-a -(a 2 _1)'2){R

-a +(a2 _1)l/2 =0

(7.22)

If we solve Eq. (7.22) with respect to R, 149

R =

1 a+(a2 -1) / 2

1 +K

1

+ (KlS)+2

2K

(7.23)

s}.3

We can obtain the Kubelka-Munk equation (7.15), if we solve this equation above with respect to KIS. To carry out qualitative and quantitative analysis using the diffuse reflectance method, we need to use an equation which links the intensity of diffuse-reflected light and an absorption characteristic of a sample. An equation that is often used is the following equation which is yielded from the Kubelka-Munk equation. K

= cosh(log(R)) -1

(7.24)

Equation (7.24) is illustrated in Fig. 7.15. Although K/S (which is related to the absorption characteristic of the sample) and log(l/R) (which is related to the intensity of diffuse-reflected light) are not in a linear relationship with each other, in a narrow KIS range we can regard that the two are approximately in a linear relationship with each other. When we analyze an infrared diffuse-reflectance spectrum, log(1/R) is usually indicated along the vertical axis. Problems with the diffuse-reflectance method are an influence by regular reflection and a scattering coefficient. The former is influenced by the size, the shape and an absorption coefficient of powder. In general, the smaller a powder diameter is, the weaker the regular

4

2

0

s

log (l/R) Fig. 7.15. Relationship between log (1/R) and K/S derived from Kubelka-Munk equation. 150

reflection light becomes. The latter is influenced by the diameter, the shape, a filling density, etc., of power. Hence, infrared diffuse-reflectance spectra are usually measured for samples mixed with KCl or KBr powder to reduce the influence of regular reflection. For a sample that interacts with KCl or KBr, diamond powder and silicon powder are also employed as a diluent. The Kubelka-Munk theory is based on a continuum model. It does not consider the particle nature of the powdered samples. The scattering coefficients is affected by packing density and particle size and hence affects the resulting infrared spectrum. Studies have been undertaken to investigate the effect of particle size, size distribution, packing density and the orientation of the sample cup [13-15]. Generally, the spectra of the same sample vary 15-30% (in absolute KM units) when run by several people [16]. Griffiths and co-workers demonstrated the effect of pressure on the diffuse reflectance infrared spectrum of powdered samples. They concluded that the duration of pressure applied is also an important factor for spectral reproducibility. They obtained a spectral reproducibility of 3% relative standard deviation after subjecting the sample to high pressure (order of tons per square inch) over a period of 15 min. The problem arising from the factors adherent to the physical nature of the sample was approached by Christy et al. [15,17] in two different ways. 1. For a sample with a narrow range of particle size distribution, the packing was carried out by an automatic packing machine to eliminate variations arising from packing pressure and time. 2. The sample cup was rotated slowly during the spectral measurements to eliminate the inhomogeneity of the sample arising from the different shapes and sizes of the sample particles. However, these techniques are not widely used because of practical problems. Attempts have also been made to modify the Kubelka-Munk function to include particle size. The Kubelka-Munk function was shown to possess an excellent inverse relationship with particle size (see Fig. 7.16) of the samples [18]: f(R)

oc lid

(7.25a)

where d is particle diameter. 151

0.37

0.30

0.23

;

0.16

i .0 0.09

0.02

3280

3120

2952

2784

Wavenumber cm' Fig. 7.16. Diffuse reflectance spectra of 4%(w/w) samples of mono-disperse polystyrene -1 spheres in KBr in the region 3300-2700 cm .

By introducing particle size in the Kubelka-Munk function (Eq. (7.25b)), quantitative determinations could be made using the diffuse reflectance technique [18]. f(R) = (KISd

(7.25b)

However, the above equation must be taken strictly as an empirical relationship because there is no theoretical justification for the inclusion of the particle size in the Kubelka-Munk function. The diffuse reflectance spectrometry is applied to a wide variety of solid samples. The samples that can be ground can be easily mixed with ground KBr powder for spectral measurement. Furthermore, samples that are dark like coal and kerogen are comfortably measured using the diffuse reflectance technique (see Fig. 7.17). Samples that are difficult to grind, for example asphaltenes can be measured using a technique used by Christy et al. [19]. In this technique the asphaltene sample is dissolved in a low boiling liquid such as dichloromethane. A background spectrum is measured 152

a

E

I0

2

e

a0 4U WU

Wavenumber cm-I Fig. 7.17. Diffuse reflectance spectra of three different coke samples with their volatile matter content (VMC) values.

with firmly packed KBr powder. Then the asphaltene solution can be carefully dropped onto the surface of the KBr powder. The sample penetrates the particles and settles on the surface of the particles. Since the diffuse reflectance technique is a near surface technique, a small amount of the sample from a few drops of the solution is more 153

. TRANSMITTANCE . SPECTRUMi

I

I

i

TRANSMITTANCE SPECTRUM

U)

z Z

40Do

I

Ii I I 3000

200 2000

I 1600

i

I

1 In 1000

6 To

v DIFFUSE REFLECTANCE SPECTRUM id

4000

A, _~~~~~~~~~~~~~~

3000

2000

1600

1000

600 1/cm

WAVE NUMBER

Fig. 7.18. Diffuse reflectance spectrum of asphaltene (obtained by deposition technique) and absorption spectrum of the same amount of sample in KBr using the transmission technique.

than enough to acquire a good quality diffuse reflectance spectrum (Fig. 7.18). The sampling technique in diffuse reflectance spectrometry has also been modified to suit high temperature and high-pressure studies. Thermal dehydration and decomposition reactions have been carried 154

out [2--23]. An example using high temperature in diffuse reflectance infrared spectrometry is given in Chapter 9.

7.4 PHOTOACOUSTIC SPECTROSCOPY When a sample is irradiated by monochromaticlight it is absorbed and converted into heat. This heat is propagated into the gas surrounding the solid and causes a variation in pressure. If the gas is contained to surround the sample, then the pressure variation can be detected as an acoustic signal. The photoacoustic spectroscopy is based on this signal. In photoacoustic spectroscopy, the sample is placed in a cell containing a gas. A microphone is attached to the cell for the detection of the photoacoustic signal. A sketch is shown in Fig. 7.19. Rosencwaig and Gersho [23] have discussed the theory of the photoacoustic effect with solids. The theory was developed considering the diffusion of heat generated by the absorption of electromagnetic radiation by a solid sample. When infrared radiation strikes the surface of a sample that has an absorption coefficient cm-l, the radiation intensity decays exponentially as I=I e

(7.26)

where I is the incident energy and I is the energy at depth x. The optical absorption depth of the sample is p = 1/P cm. If the radiation is

Zig

CGas Microphone / /

I' /

// zzr

F=-o+lb) Fig. 7.19. Sketch of the setup used in photoacoustic measurements. 155

modulated sinusoidally at o rad s-l, then the intensity of radiation at a depth x is (7.27)

I = Io (1 + cosot) e-

Absorption of infrared radiation at a depth x produces heat energy PI per unit volume (heat density). PI = X PIo (1 + cosot) e- >

(7.28)

The heat generated then decays exponentially as (7.29)

PI e

where a is the thermal diffusion coefficient. The heat decay can then be written as

4 1Io (1 + cost)

e

e - = X pIo (1 + cosot) e- (P+"'

X

(7.30)

The diffusion depth of the heat generated is p, = 1/a cm. The diffusion depth can be related to the physical parameters Ks, p,, Cs as Ps = (2KJpsCsc o)

(7.31)

where Ks is the thermal conductivity, p, is the density, and Cs is the specific heat of the sample. The oscillating heat is then dissipated to the gas that is in contact with the solid. Equation (7.31) shows that the heat generated at depth x is oscillating. By considering the sample thickness from x = 0 to x = -(l + lb) and gas column length from x = 0 to x = +lg, Rosencwaig and Gersho [23] have solved the thermal diffusion equation for the system, temperature distribution in the cell, the periodic heat variation in the gas and the acoustic pressure wave produced in the cell. The equations are too complex to deal with here. However, the solution for the incremental change in pressure in the cell that is responsible for the production of acoustic signal is shown in Eq. (7.32). The term Q varies with the opaqueness of the solid sample and the gas in the cell. 6P(t) = Q e- [ (wt -h/4)] 156

(7.32)

Rosencwaig and Gersho [23] have divided the solids into six different categories depending on their optical opaqueness to explain the dependency of Q and the signal: three cases for optically transparent and three cases for optically opaque solids according to their relative magnitude of the thermal diffusion length (p,), as compared to the thickness of the solid () and optical absorptionlength (p). (a) For optically transparentsolids where

pa

> 1, the cases are

1. Thermally thin solids where p, >> 1 and p, > p. 2. Thermally thin solids where ps >> 1 and p < pp. 3. Thermally thick solids where p. < 1 and yp 1 and p, >> pp. 2. Thermally thick solids where ps < 1 and pi > pp. 3. Thermally thick solids where p, 10x the shortest scan period of the conventional continuous-scan interferometer, each interferogram can be considered to be instantaneous. However, even though scan rates of > 50 Hz have recently been achieved on commercial instruments (using bi-directional scanning), this is always at the expense of resolution and, in any event, can only be applied for a minimum of -20 ms time resolution. In addition, this is a technique limited strictly to the impulse/response mode. Another continuous scan approach, which is applicable to synchronous modulation experiments is to scan the mirror slowly enough that the highest Fourier frequency generated in the spectral bandwidth of interest is more than 10x lower than the external modulation applied to the sample. This method has been successful, but it requires an interferometer of exceptional stability [34]. Even so, it is not practical for external modulation frequencies of < 400 Hz, except in the far-IR. 8.1.8 Step-scan dynamic FT-IR As an alternative to the continuous scan mode of interferometry, the data may be collected in the step-scan mode, in which the retardation is changed incrementally and data are collected while the retardation is held constant. As previously stated, step-scan interferometry is more universal in its application than is the continuous-scan method since it can be applied without fundamental restrictions, to either synchronous modulation or impulse/response experiments. Since the retardation is constant while data are collected (or, as in some cases, the retardation is modulated about a fixed valve), the spectral multiplexing is uncoupled from the time-domain. Furthermore, step-scan FT-IR offers both conceptual and practical simplicity since the experimental parameters can generally be changed independently and with ease. Although the step-scan mode of interferometry predates the continuous-scan mode by decades, until recently it has not been widely available to experimenters in a form suitable for routine use outside the far-IR [28]. Both types of time-resolved experiment, either impulseresponse or synchronous modulation experiments, can be performed 209

using the step-scan mode of operation. For impulse/response experiments data are collected as an explicit function of time at the desired intervals after each impulse. The perturbation/impulse is repeated at each step as many times as necessary to achieve the desired signal to noise. Data from all times are sorted by time and transformed to produce the time-resolved spectra. 8.1.9 Step-scan impulse-response experiments The development during the last decade of modern step-scan interferometry instrumentation has allowed FT-IR to be applied to the study of time-dependent phenomena in ways not previously possible, because of the problems of uncoupling the spectral multiplexing from the temporal domain in the continuous-scan FT-IR mode [35]. Specifically, the time regime from tens of nanoseconds to tens of milliseconds has been accessible to time-domain measurements to only a very limited degree with continuous-scan instrumentation and not at all for modulationdemodulation (frequency-domain) experiments in this time range. The step-scan technique not only works very well in this time regime and for slower phenomena, but is only prevented from application to faster processes by the signal strength, the speed of available detectors, the intensity of sources, and the speed and sophistication of the electronics. As in the synchronous modulation experiments, it is necessary that the response of the system to the perturbation in the impulse-response time-resolved, or "time-domain" mode should be perfectly reversible, so that any desired number of repetitive pulses can be used to achieve the necessary signal averaging. In these experiments a signal from the spectrometer externally triggers the pulse generator, which then produces a voltage step across the cell and maintains this voltage for a time along with respect to the response time of the sample. As stated above, repetitive pulses (separated by a suitable recovery interval) at each interferometer position are used for signal averaging. The resulting data are then sorted by time to produce individual interferograms for each time t, which are then transformed to give the time-resolved spectra. Figure 8.6 shows the data collection scheme for the simplified case of one excitation pulse for each retardation step. The data are sorted vertically to produce the time-resolved interferograms. The time intervals are usually equal, but this is only an experimental convenience, not a fundamental requirement. 210

Retardation Time Intervals 60

to*, tl,

t2,

1

to*, t,

t2

$n

,

.

to*, tl, t2, ·.

.

.....tn

.

., tn

. .,

tn

Fig. 8.6. Data collection scheme for step-scan impulse-response experiment.

An example of the application of a infrared time resolved technique will briefly take place here. In this study, a time-resolved infrared spectrometer in the spectral region of 700-4000 cm-l was constructed with a resolving power of 50 ns and detection limit of 10-6 [36]. Isomerization of retinal by light irradiation was examined to suggest an isomerization mechanism from trans form to 3:1 mixture of 13-cis and 9-cis forms within < 50 ns. The time-resolved IR spectra of N,Ndimethylamino-4-benzonitrile in polar BuOH and nonpolar hexadecane with and without oxygen bubbling gave conclusions that the 2096 cm-l band appearing only in polar solvent with a life of several ns and no oxygen effect was due to CN stretching vibration of the excited singlet state of twisted intramolecular charge transfer structure. A 2040 cm-l band appearing also in polar solvents with longer life in the order of 100 ns and disappearing by oxygen bubbling was due to CN stretching vibration of the excited triplet state.

8.2 APPLICATIONS TO LIQUID CRYSTALS AND LIQUID CRYSTAL POLYMERS Thermotropic liquid crystal polymers have awakened a great deal of interest in the past decade from both technological and scientific points of view. Great emphasis was placed on the modification of physical and thermal properties, and analysis of the corresponding structureproperty relations [37]. Vibrational spectroscopy is a useful tool in the characterization of these materials, and the type of information available through infrared spectroscopy relates to crystallinity and polymorphism, phase transitions and orientational behaviour. 211

Efforts to investigate the responses of liquid crystals to applied electric field using vibrational spectroscopy started in 1981, using attenuated total internal reflectance (ATR) spectroscopy [38]. In addition, the time course of the reorientation is most suitably studied by the use of dynamic vibrational spectroscopy. Both time-resolved Raman spectroscopy [391 and dynamic infrared spectroscopy [40] have been applied to the electric-field induced reorientation of nematic liquid crystals. Coles and Tipping used microsecond time-resolved Raman spectra of a nematic liquid crystal as a function of the applied electric field [41]. Kaito et al. used fast FT-IR scanning with millisecond time resolution to study the time-dependent polarized infrared absorption of a nematic liquid crystal [42]. Toriumi et al. published the first stroboscopic FT-IR data with sub-millisecond time resolution in 1988 [43]. In 1991, Gregoriou et al. published the first synchronous modulation data on the reorientational behaviour of a nematic liquid crystal in response to an AC electric field [44].

8.2.1 Dynamic IR spectroscopy of polymers Excellent reviews of the relevant instrumental and theoretical background of polymer deformation and relaxation studies by simultaneous Fourier-transform IR spectroscopic and mechanical measurements exist in the literature [45,461. In the first review, the vibrational spectroscopy of stressed polymers, orientational measurements using infrared dichroism, deuteration, and experimental results for thermoplastics and elastomers are discussed. Dynamic rheo-optical IR techniques promise an exiting future for vibrational spectroscopy as a tool on polymer research [47]. The earlier work by Noda et al. [48] has been successfully adapted to interferometric measurements using step-scan FT-IR techniques [49]. Noda et al. [50] performed a dynamic infrared linear dichroism study of high density and low-density polyethylene films near the -transition temperature. It was found that a different deformation mechanism operates above and below T. Specifically, a negative dynamic dichroism at 1473 cm7l and a positive dynamic dichroism at 1463 cm -l were observed at 32°C. This observation was interpreted to mean that above T, the orthorhombic crystallographic b axis reorients parallel to the direction of applied strain, while the crystallographic a axis reorients perpendicular to the strain direction for both HDPE and 212

LDPE. The dynamic dichroism of both bands changes sign at -50°C (below Tp), indicating that the dynamic reorientation directions for the crystalline a and b axes are shifted below To. In this earlier work with dispersive instrumentation switching between spectral regions was a difficult task, requiring a substantial investment in time. Lefebre et al. have performed infrared measurements on the PS/ PPO compatible blend in terms of static uniaxial strain above the glass transition temperature [51]. Evidences that the PPO and PS chains orient in a different way were found, in spite of the compatible nature of the blend. The PPO orientational behaviour does not depend on PPO concentration in the concentration range that was studied (0-35%) while PS orientation regularly increased up to 25% PPO and then remained constant. The authors suggested two explanations for this behaviour. Either the increase of the PPO concentration results in an increase of the knots of the physical network, or the relaxation of the PS chains is hindered by PPO chains. The first explanation is not supported by their experimental results, since the PPO orientation remained constant as the concentration increased. In a similar study, side chain liquid crystalline polyurethanes are a new class of materials that show promise for mechano-optic applications. The rich morphology afforded by these materials also provided a chance to understand the interplay between polyurethane morphology and liquid crystalline ordering. In this study, the response of a polyurethane with liquid crystals pendant to the soft segments to an applied strain using Fourier Transform Infrared (FT-IR) linear dichroism was detected. It was found that this complex material followed trends established in the literature for both side chain liquid crystalline homopolymers and segmented polyurethanes. At low strains, the soft segments aligned with strain inducing an orientation in 'lone' hard segments. Up to strains of 40%, the LC mesogens align with the strain field and the hard segments in hydrogen bonded domains align perpendicular to the field. At strains above 40%, a rearrangement of the ordering was found that resulted in the smectic layers and the hard segments aligning parallel to the field. In addition, dynamic FT-IR experiments showed that the viscoelastic reorientation of various segments of the macromolecule could be monitored as a function of the applied strain. For the polyurethane under study, the cyano band was used to follow mesogen movements, and the urethane carbonyl to track the hard segment. Evidence were presented for two types of hard segments: those involved in hydrogen bonding within 213

hard domains, and those found in 'lone' hard segments in the soft matrix. Evidence were also found for two types of mesogens: those found in smectic layers, and those not involved in smectic ordering at the hard domain interface. The hard domains and the smectic layers had strong viscous components to their mechanical response. The 'free' mesogens and the 'lone' hard segments, on the other hand, exhibited a more elastic response. A model was proposed to represent these findings, and reflections on the cooperative movement of the different macromolecular components of the polyurethane were offered [52,53].

8.3 APPLICATIONS TO OPTICALLY ACTIVE MATERIALS 8.3.1 Organic light emitting diodes (OLEDs) Electroluminescentdevices based on organic low molecular weight (e.g., Alq 3) and polymeric materials (e.g., PPV) are recently attracting much attention mainly due to applications as large area light-emitting devices (OLEDs). Generally, these devices are thin-film single-layer or multilayer structures composed of a hole transport and an emitting and an electron transport material sandwiched between two electrodes. OLED devices are generally fabricated utilizing vapour deposition techniques or film-casting techniques from solution. The structure and the correlation between intermolecular interactions and optical properties in various such systems have been investigated with infrared spectroscopy among other methods. Some selected examples will be presented here. In one such study, the authors reported the presence of carbonyl moieties as defects, formed during the thermal conversion of a precursor to poly(p-phenylenevinylene) (PPV) [54]. The increase in carbonyl groups was correlated with a dramatic reduction of PPV photoluminescence. If the conversion is carried out in a reducing atmosphere, e.g., 15% hydrogen in nitrogen, and the amount of carbonyl moiety was substantially reduced and the photoluminescence intensity of the polymer increased as much as five-fold. In another study, thin films of a cross-linkable hole-conducting monomeric triarylamine for use in organic light emitting diodes were examined [55]. The rate of photo-crosslinking and the overall polymerization yields were measured using real-time FTIR spectroscopy. The electronic properties were characterized in a typical diode configura214

OCH,,

OC.H, OC,H,

H,C,

CH,

_0

H,,Co

O,

OOo O

H,,C,O

H,C,O 0

"'v" OC,H,

c

o

2

rCH, 0

OH,, OC

0

H'C•ka---o

OC,H,,

OC,H,

,

XCH,,

H,,C,O.

OH"CO 'a i' H,C,

o-"

2b

NC

"O- '

o ON

3

C

OC,H,,

'

H,

OCH,,

HC

0/

"

_

0

0

H~C'-o O

*CO-'

0CM

-

M-~

c 0

H,,CCO

H'C-O

OCH,

0

OC, H ,,

Fig. 8.7. Chemical structure of the mono-1, bis-2a-c and tris acrylate 3,4 derivatives of hexaalkoxytriphenylenes. (Reproduced from Ref. [56] with permission. Copyright (1999) American Chemical Society.)

tion using InSn oxide and Al as the contacts. The current passed through the devices was limited by the injection of holes into the semiconducting polymer layer by tunnelling. Cross-linked layers withstood approximately 20 times higher currents than non-cross-linked layers before dielectric breakdown occurred. In a similar study, new hexaalkoxytriphenylenes having one, two, or even three lateral attached acrylate moieties as polymerizable groups were synthesized and characterized for use as novel insoluble hole transport materials in organic LEDs [56]. Figure 8.7 shows the chemical structure of the mono-1, bis-2a-c and tris acrylate 3,4 derivatives of hexaalkoxytriphenylenes. The conditions for the photopolymerization of these monomers in thin film were evaluated and tested. The bisacrylates and trisacrylateswere used to build insoluble networks. When a mask was used during the irradiation, patterned films were prepared. The polymeric reaction was controlled by GPC and FTIR spectroscopy. Figure 8.8 shows the spectroscopic data that verify the 215

0 co o

al

1C

wave numbers [cm'] Fig. 8.8. Spectroscopic verification of the photo-cross-linking reaction in thin films by FTIR (16 scans; 4 cm - ' resolution, films on KBr): (a) trisacrylate 4 before polymerization; (b)photopolymerized film of 4; (c) reference spectrum of a polymer derived from solution polymerization of 1. (Reproduced from Ref. [56] with permission. Copyright (1999) American Chemical Society.)

photo-cross-linking reaction in thin films. The networks and patterned structure were also confirmed by UV spectroscopy, surface profiles, and SEM photographs. Since hexaalkoxytriphenylenes are known as excellent photoconductors, the photopolymerized films were used as hole transport layers in two layered OLED with Alq3 . Finally, Figure 8.9 illustrates the OLED characteristic of a two-layer device with Alq3 as emitting/electron transport layer and a cross-linked film as a holetransportlayer. Polymer light-emitting diodes, based for example on MEH-PPV, are known to be susceptible to oxidative degradation [571. This leads to loss of conjugation, i.e. lower carrier mobility and higher operating voltage, and to the formation of carbonyl species, i.e. to luminescence quenching. In-situ FTIR revealed that ITO can act as the source of O. To explore further the mechanism of oxidation and to provide guidance for its elimination, the authors have studied the behaviour of MEH-PPV LEDs prepared with a variety of conducting polymer anodes including polyaniline and polythiophene derivatives cast from various solvents and with various molecular and polymeric dopants. In all the cases examined, polymer anodes led to significant improvement in lifetime over devices with ITO as the anode contact. Also, in contrast to the variability observed for ITO anodes, conducting polymers with 216

voltage [V] -15

-10

-5

0

5

10

15

20

25

10' OTo

E '

10'

C lo" L ,

2 10

electric field [10 e V/cm] Fig. 8.9. LED characteristic of a two-layer device with Alq3 as emitting/electron transport layer and a cross-linked film of 3 as hole-transport layer ITO/3 (35 nm)/Alq 3 (35 nm)/Al (200 nm). (Reproduced from Ref. [56] with permission. Copyright (1999) American Chemical Society.)

polymeric dopants yield consistently good devices with power efficiencies of approximately 0.5% at 5 V and brightness >1000 cd/m 2 . Anodes prepared with small molecular dopants are more variable and exhibit short-term behaviour which suggested interfacial electrochemistry. The authors described the device characteristics in the context of a model of hole-dominated bipolar charge injection with Langevin recombination. In another study, poly(p-phenylenevinylene) (PPV) was derived from a sulphonium salt precursor by ion beam irradiation [58]. A quadrupole mass spectrometry analysis of the evolved species showed rapid loss of HC1 and tetrahydrothiophene groups during irradiation with 100 keV Ne +, indicating precursor degradation. Rutherford backscattering spectrometry confirmed the reduction of the sulphur and chlorine content in the PPV film, whereas infrared spectroscopy showed that the vibration mode at 2940 cm - l for the sulphonium group has vanished for a 2 x 1016 ion effluence. The appearance of the transvinylene peaks, at 3024 and 965 cm-l in PPV indicated the full conversion of the precursor into the conjugated polymer for this effluence. The correlation between a narrower optical band gap and the by one order of magnitude higher conduction of a film implanted with Na + ions with respect to a Ne + irradiation showed the doping effect induced by an implantation with electronically active species. 217

8.3.2 Conducting polymers In the area of the application of FT-IR techniques to study conducting polymers, excellent reviews on the theory of polarons, bipolarons, and solitons and use of vibrational spectroscopy in the studies of selflocalized excitations and charge transfer in conducting polymers exist [59]. Geometry of polymer chains and changes induced by doping; electronic states of polarons, bipolarons, and solitons; electronic absorption spectra of poly(p-phenylene) and detection of radical anions in p-oligophenyls were some of the issues addressed. In addition, the first vibrational spectroscopic investigation of a novel non-aqueous proton conducting polymer gel electrolyte consisting of a PMMA matrix and a solvent mixture (ethylene carbonate (EC)/ propylene carbonate (PC) or EC/PC/N,N-dimethylformamide (DMF)) with a dissolved organic acid (benzoic or salicylic acid) took place [60]. The protonic conductivity of the gels was of the order 10-4-10 - 5 S/cm at room temperature. It was found that the conductivity was proportional to the degree of dissociation of the acid, the latter determined from Raman spectroscopic data, and that the degree of dissociation depended on the properties of the solvent mixture. Finally, the relationship between the proton conductivity and the solvent-diffusion dynamics was also studied. Electronic absorption and vibrational spectroscopies of doped conjugated polymers, whose ground states are nondegenerate were also studied [61]. These studies have concluded that polarons are the major species generated by doping in most nondegenerate conjugated polymers such as polythiophene, poly(p-phenylene), and poly(p-phenylenevinylene), in contrast with the previous view that bipolarons are the major species. 8.3.3 Composites and nanocomposites FT-IR spectroscopy has also been used in the study of composite materials and nanocomposite films. In one such example, infrared spectroscopy was used to monitor the formation of thin films of photosensitive hybrid organic-inorganic glass on silicon via the solution sol-gel method [62]. Glasses consisted of photoinitiator,methacryloxypropyltrimethoxysilane, methacrylic acid, and zirconium oxide. Clear, low optical loss films were obtained, indicating nanophase homogeneity in the samples. The nanocomposite films were suitable for 218

fabricating optical components such as ridge waveguides and Bragg diffraction gratings. The increase in the refractive index of the glass relative to the surrounding material during photolithographic processing was identified as a key material parameter in device fabrication. Accordingly, electronic and vibrational spectroscopy were used to provide insight into the structural changes that occur when glasses were irradiated with continuous narrow band 4.9 eV and pulsed 6.4 eV light. Arguments were advanced, linking the changes in refractive index to collateral densification leading to volume compaction of the silicate network during organic free-radical polymerization. This was shown by following the time evolution of relevant infrared absorption bands. Free silanol and unreacted methoxysilane were consumed in the process. Matrix densification was indicated by shifts to lower wavenumbers in the transverse optical phonon mode associated with decreasing Si-O-Si bond angles of the asymmetric stretching vibration. Growth in the Si-O-Si framework was observed through increased intensity in this infrared absorption. Similar behaviour was observed for films irradiated with 6.4 eV light from an excimer laser. A phase mask in combination with pulsed 6.4 eV light was used to inscribe a 1.5 mm, high-reflectivity polarization-independent Bragg grating into a ridge waveguide. The high reflectivity is thought to arise from a periodic modulation of the volume compaction of the matrix. Overall, the organic component of the glass confers unique properties on the material that allows it to be densified even with 4.9 eV light. By comparison, sol-gel silica with no organic component must be densified at nearly twice the photon energy. Di(carboxystyryl)benzene was self-assembled with a Zn complex in THF on SiO2 and Si substrates to form thin films that exhibited bluegreen luminescence [63]. FT-IR spectroscopy of a 56 nm film on Si showed characteristic absorption bands at 1600 cm-l, 1543 cm-l, and 1412 cm-l consistent with a powder sample. The refractive index (n) was 1.66 at 633 nm. Multilayer growth proceeded by a 15 Angstrom increase after initial surface coverage. These films were pursued for the preparation of self-assembled films for electroluminescence applications. In a similar study, high-energy milling provided an effective and environmentally conscious method for nanosizing Si [64]. Colloidal suspensions of nano-sized Si were demonstrated and used for the fabrication of high refractive index nanocomposites. Si nanoparticles with average sizes of 20-40 nm and size distributions of approximately 219

25% were separated from milled powder via sonication and centrifugation. These nanoparticles were analyzed using TEM, dynamic light scattering, x-ray diffraction and UV-visible/FTIR spectroscopy. Formation of stable colloids was in part attributed to a thin surfaceoxide layer. The decrease in the average particle size caused a blue shift in their absorption spectrum, thus increased the transparency in the red part of the visible region. These Si nanoparticles were used to fabricate high refractive index nanocomposites, with refractive indexes

I

To To

Wavenumber (cm-')

t I

Fig. 8.15. Series of spectra acquired from a 300 mm wide region of normal human trabecular bone. (Reproduced from Ref. [78] with permission. Copyright (2000) Society for Applied Spectroscopy.)

tissue. Finally, Figure 8.16 shows the infrared images of the index of mineral crystallinity/perfectionand a histogram of this quantity for an estrogen treated site in a fractured rat femur and for an untreated site. Fourier-transform IR microspectroscopy was used to study bone mineralization processes in an in vivo model and in enamel in osteogenesis imperfecta [78]. The ability of this technique to map new bone formed in implanted macroporous calcium phosphate biomaterial from sections was reported for the first time. This technique allowed the correlation of the microstructure of bone formation in the in vivo model with modifications in carbonate and phosphate environments of the mineral phases during maturation. Analysis on enamel sections revealed changes in the mineral environment of carbonate and phosphate ions and probably in the size of the enamel crystals. These modifications contributed to the fragility of enamel in osteogenesis imperfecta. The infrared functional group imaging of a part of the implanted biomaterial and the bone ingrowth provided the visualization of chemical modifications occurring in biomaterial implants at 20 pm spatial resolution. The use of this technique, in conjunction with appropriate sampling methods and data analysis should provide further insight into the molecular structure of mineral phases of calcified 229

·--

119m:I

11

CD

.4 r.

To

:

Z

=

I 00

z;

'Y'"

,

I 11 .2

111

1(030)/1( 1020)

400 pm ,~K ii-

10

$s

E

.40 0 10

40

12 I 12

C) I

-

1(030)/1( 1020) 400 pm

Fig. 8.16. Infrared images of the index of mineral crystallinity/perfection and a histogram of this quantity for an estrogen treated site in a fractured rat femur (bottom) and for an untreated site (top). (Reproduced from Ref. [78] with permission. Copyright (2000) Society for Applied Spectroscopy.)

tissues and help to elucidate mineralization processes, skeletal disorders and properties of the biomaterials used as bone substitute. The distribution of chemical species and the degree of orientation in semicrystalline polymer systems have also been studied using fast Fourier transform imaging [79]. A variety of poly(ethylene glycol) systems, including pure polymer, high- and low-molecular weight blends, and blends with amorphous polymers, were studied. It is shown that fast FTIR imaging can be used to determine the distribution of species with different molecular weights and can be used to determine the degree of segregation of different components in blends with amorphous polymers. Additionally, by employing an infrared polarizer, 230

the degree of orientation was determined in these systems by the generation of spatially-resolved dichroic ratio images. Imaging spectrometry enables passive, stand-off detection and analysis of the chemical compounds of gas plumes and surfaces over wide geographic areas [80]. The authors described the use of a longwavelength infrared imaging spectroradiometer, comprised of a loworder tunable Fabry-Perot etalon coupled to a HgCdTe detector array, to perform multispectral detection of chemical vapour plumes. The tunable Fabry-Perot etalon used in this research provides coverage of the 9 . 5 -1 4 -pm spectral region with a resolution of 7-9 cm-l. The etalonbased imaging system provided the opportunity to image a scene at only those wavelengths needed for chemical species identification and quantification and thereby minimized the data volume necessary for selective species detection. The authors present initial results using a brassboard imaging system for stand-off detection and quantification of chemical vapour plumes against near-ambient temperature backgrounds. Model calculations were presented comparing the measured sensitivity of the sensor to the anticipated signal levels for two chemical release scenarios. In another study, a 64x64 Mercury-Cadmium-Telluride (MCT) focal-plane array detector attached to an FT-IR microscope was used to spectroscopically image 8-pm-thick cross-sections of wheat kernels in the fingerprint region of the IR spectrum [81]. After fast-Fourier transformation of the raw image interferograms, the data can be displayed either as a series of spectroscopic images collected at individual wavelengths, or as a collection of infrared spectra obtained at each pixel position in the image. Image contrast is achieved due to the intrinsic chemical nature of the sample at each pixel location in the image. Individual cell layers near the outer portion of the wheat kernel, as well as the primary root within the germ, can be clearly differentiated in the IR images as a result of this enhanced chemical contrast. Micro-imaging spectrometers incorporating focal plane array (FPA) detection require careful demarcation of cold shield aperture size for both optimal performance and prevention of errors. One study explored the effects of changing the diameter of the cold shield aperture on the intensity and spatial homogeneity of the incident radiation [82]. A uniform polystyrene film was repeatedly imaged by using cold shields of varying aperture sizes. It was shown that a smaller than optimal aperture size led to image edge clipping, resulting in an inefficient use of the array, lower overall signal, spectral distortions, and higher noise 231

characteristics. Use of an aperture size larger than required caused a decrease in the effective dynamic range of measurements, resulting in higher noise levels. The advantages and necessity of optimizing imaging spectrometer performance by employing a cold shield with an appropriately sized aperture were discussed. The penetration of chemical reagents through human hair after bleaching has been spatially characterized using IR microspectroscopy with a synchrotron source [83]. Chemical imaging of hair cross-sections before and after bleaching was achieved with high contrast, using the peptide and lipid mid-infrared absorption bands which are characteristic of hair. The ability to make images using functional groups as a contrast mechanism can be applied to studies of other chemical groups, if present, in the structure of the hair. In this study it was shown how the penetration of an organically active reagent in the hair structure could be quantified with a spatial resolution of few microns. These results demonstrated that synchrotron infrared microscopy is a powerful tool for characterizing chemical interactions of hair samples with specific cosmetic materials. Infrared spectra of breast tumour cell lines and breast tumour tissues have been measured. Infrared measurements of tumour cells revealed that approximately fifteen cells are necessary to obtain spectra of good signal-to-noise ratio using an IR microspectrometer equipped with a conventional IR thermal source [84]. Comparative studies of human breast tumour cell line suspensions demonstrated that MCF-7 cells and drug-resistant NCI/ADR cells could be differentiated based on their infrared signatures. The most striking differences between MCF-7 and NCI/ADR were found in features assigned to CH2 and CH3 stretching vibrations of lipid acyl chains and PO,- stretching vibrations of nucleic acids. To assess the potential of IR spectroscopy for the diagnosis of breast tumour tissues, thin sections of tissue were mapped by FTIR microspectroscopy. The spectra of these maps were analyzed using functional group mapping techniques and cluster analysis and the output values of the different approaches were then reassembled into IR images of the tissues. A comparison of the infrared images with the standard light microscopic images of the corresponding areas suggested that: (i) chemical mapping based on single band intensities was an easy way to detect microscopic fat droplets within tissue; (ii) the comparison of IR images based on band intensities at 1054 and 1339 cm-l provided information on tissue areas containing tumour cells; and (iii) cluster analysis of the spectra was superior to the 232

single band approach and more appropriate for differentiation between tissue types. In an different approach, using synchrotron radiation as an ultrabright infrared source, the authors were able to map the distributions of functional groups such as proteins, lipids, and nucleic acids inside a single living cell with a spatial resolution of a few microns [85]. In particular, the changes in the lipid and protein distributions in both the final stages of cell division and also during necrosis were mapped. FT-IR microspectroscopic maps of unstained thin sections from human melanoma and colon carcinoma tissues were obtained on a conventional IR microscope equipped with an automatic x, y stage [86]. Mapped infrared data were analyzed by different image re-assembling techniques, namely functional group mapping ('chemical mapping') and, for the first time by cluster analysis, principal component analysis and artificial neural networks. The output values of the different classifiers were recombined with the original spatial information to construct images whose colour or grey tones were based on the spatial distribution of individual spectral patterns. While the functional group mapping technique could not reliably differentiate between the different tissue regions, the approach based on pattern recognition yielded images with a high contrast that confirmed standard histopathological techniques. The new technique turned out to be particularly helpful to improve discrimination between different types of tissue structures in general, and to increase image contrast between normal and cancerous regions of a given tissue sample. Infrared absorption spectroscopy was used with other scattering and imaging techniques to elucidate the interface reactions leading to permanent chemical bonding ofjoined hydrophilic wafers upon annealing, and to uncover the thermal evolution of H-decorated defects in Himplanted Si wafers [87]. Detailed mechanisms were proposed whereby the role of micro-voids as gathering sites for H2 is highlighted and the kinetic interplay between defect formation/evolution, H passivation of internal structures and molecular H2 formation was critical for exfoliation to occur. The demand also of smaller device dimensions drives the need to improve the lithographic and the metrological tools to produce them [88]. Characterization of the image formation during the lithographic process is key to any process control effort. Scanning probe microscopy (SPM) on exposed, unbaked and baked, undeveloped photoresist showed morphological details of the image formation process unachievable 233

with other techniques. The use of micro-FT-IR spectroscopy was investigated for latent image chemical analysis. Both of these techniques were used in the study of the dependence of the latent image of a negative novolac-based chemically amplified resist, SAL 605 by Shipley, with post-exposure bake (PEB) conditions. The objective of the experiment was to understand how the thermal properties of the resist and the linking reaction taking place were related to each other during PEB. Experimental results indicated that resist from unexposed regions diffused into the exposed resist during PEB. SPM results show that this diffusion increased as the PEB temperature rose above the oxide glass transition temperature of the unexposed resist. These results showed that the linker component of the resist, hexamethoxymethylmelamine, was identified as one of the resist components that diffused into the exposed regions during PEB. Traditional methods of cell wall analysis have provided valuable information on wall composition and architecture, but, by having to rely on the use of bulk samples, have averaged out this intrinsic heterogeneity. FTIR microspectroscopy addresses this problem by providing chemical information from an area as small as 10x 10 pm of a single cell wall fragment or area of a tissue section that has been imaged with a microscope accessory. The authors have used FTIR microspectroscopy as a powerful and extremely rapid assay for wall components and putative cross-links. The spectra were sensitive to polymer conformation, and the use of polarizers in the microscope accessory allowed the orientation of particular functional groups to be determined, with respect to the long axis of elongating cells. The spectra constituted species and tissue-specific 'finger-prints', and the use of classical discriminant analysis may provide the opportunity for correlating spectral features with chemical, architectural or rheological wall properties. Spectral mapping of an area of a specimen allowed the morphological features resulting from cell growth and differentiation to be characterized chemically at the single cell level. In addition, the fidelity of the spectral images was determined by the pixel number of the focal-plane array [89]. In another study, an instrument was described that simultaneously recorded images and spectra of materials in the infrared fingerprint region using a long-wavelength infrared focal-plane array detector, a step-scan Michelson interferometer, and an IR microscope [90]. With the combination of step-scan Fourier transform Michelson interferometry and arsenic-doped silicon Si:As focal-plane array image 234

detection, an infrared spectroscopic imaging system was constructed that maintained both an instrumental multiplex and multichannel advantage and operates from approximately 4000 to 400 cm-l. With this method of mid-infrared spectroscopic imaging, the fidelity of the generated spectral images recorded through the microscope was solely determined by the number of pixels on the focal-plane array detector, and only a few seconds of data acquisition time were required for spectral image acquisition. This seamless combination of spectroscopy for molecular analysis and the power of visualization represented the future of infrared microscopy. 8.5 APPLICATIONS TO INDUSTRIAL PROCESS Recently, a new method for simultaneous determination of vulcanized rubber additives by FT-IR using partial least-squares regression for multivariate calibration was developed [91]. The effect of various wavenumber ranges and the use of the absorbance and first-derivative spectral modes on performance were studied by applying the method to three different sample batches containing several additives in different proportions, all of which were resolved with satisfactory results. In another study, a simpler spectrometer design was employed for industrial process. A Michelson type interferometer was used where the moving mirror was suspended by two fluxes and driven by a coil actuator [92]. Displacement of the mirror was monitored using a much smaller transducer with a better thermal stability than the conventionally used HeNe laser. The beamsplitter is a CaF 2/Si and a thermoelectrically cooled PbSe is used as the detector. The spectral range was from 5000 to 1800 cm-1 with resolution better than 8 cm -l . Furthermore, the applications of FT-IR spectroscopy to industrial processes was greatly benefited by the use of mid-infrared optical fibres [93]. The ability to make measurements at a remote site or as a reaction occurs offers a significant advance in these types of analyses. Midinfrared fibres are used to transmit radiation outside of the spectrometer, to the sample, and then to the detector. A section of the fibre, with the protective cladding removed, is used as the sampling device. In this decladded region the fibre, acting as an internal reflectance element, contacts the sample and provides the chemical information for analysis. In one example, the use of a fibre to monitor the progress of a curing reaction in thermoset composite materials where the fibre was imbedded in the matrix was studied. 235

Chalcogenideglass fibres based on sulphide, selenide, telluride and their rare earth doped compounds are being actively investigated worldwide [94]. Great strides have been made in reducing optical losses using improved chemical purification techniques, but further improvements are needed in both purification and fiberization technology to attain the theoretical optical losses. Despite these problems, current single-mode and multimode chalcogenide glass fibres are enabling numerous applications. Some of these applications include laser power delivery, chemical sensing, imaging, scanning near field microscopy/ spectroscopy, fibre infrared sources/lasers, amplifiers and optical switches. 8.6 APPLICATIONS TO ORGANIC THIN FILMS In recent years, functional organic thin films have received keen interest in the field of molecular electronics because of an increasing awareness that functional organic thin films exhibit a variety of interesting functions [95-97]. A number of molecular devices have been proposed that are based on organic thin films such as LangmuirBlodgett (LB) films with nonlinear optical properties, photovoltaic cells, piezoelectric and pyroelectric devices, resistance and conducting materials, and chemical and biological sensors. To sufficiently understand the functions of functional organic thin films, it is necessary to investigate the arrangement and orientation of molecules in a film, and further, structures (e.g., conformations, chemical bonds, intermolecular interactions, electronic states) and the like. In addition, knowledge about the relationship between functions and structures is essential to designing of a new organic thin film. While methods of exploring the structure of an organic thin film include x-ray diffraction, atomic force microscopy, ultraviolet and visible spectroscopy, fluorescence spectroscopy, ESR, infrared spectroscopy, Raman spectroscopy, etc., the infrared spectroscopy we describe in this section can be said to be prominent in terms of the diversity and quantity of information and the simplicity of measurement [95-99]. 8.6.1 Infrared spectra of Langmuir-Blodgett films We will mainly describe here infrared studies on LB films which are attracting the greatest attention among a variety of organic thin films. In the following, LB films such as those shown in Fig. 8.17a and b will 236

(a) Y-type

(b) Y'-type Fig. 8.17. Structure of (a) Y-film and (b) Y'-film.

237

C n H 2n+1

n=12; Dodecyl-TCNQ n=15; Pentadecyl-TCNQ n=18; Octadecyl-TCNQ Fig. 8.18. Structure of 2-alkyl-7,7,8,8-tetracyanoquinodimethane.

be referred to as the Y-film and the Y'-film, respectively. When used for structural investigations on LB films, infrared spectroscopy is advantageous in the following points. 1. It is possible to measure a spectrum non-destructively at room temperature under a normal pressure. 2. Operations for measurement of spectra are relatively easy. 3. It is possible to measure a spectrum of even a one-layer LB film. 4. Since an infrared spectrum can be measured for an LB film, a solution, a solid and a crystal, one can compare a structure of a sample in the LB film with structures of the sample in other states. 5. Various types of infrared measurement methods (transmission method, the ATR method (Section 7.2.1), the RA method (Section 7.2.4), a surface-enhanced method, etc.) may be applied. As a vibrational spectrum sensitively reflects the arrangement of atomic nuclei and nature of chemical bonds within a molecule, or an interaction between the molecule and a surrounding environment it is suitable very much to study the molecular aggregation, orientation, and structure in an LB film. Knowledge obtained from infrared spectra of LB films are summarized as follows. 1. Orientation of molecules; whether hydrocarbon chains and chromophores are perpendicular to a substrate or tilted with respect to the substrate normal, etc. It is also possible to quantitatively estimate a tilt angle [100-102]. 2. Sub-cell packing of hydrocarbon chains [103,104]. 3. Conformations of hydrocarbon chains; whether hydrocarbon chains have trans-zigzag structure or partially contain gauche forms, etc. [105,106]. 238

4. Structures of chromophores; the conformation, chemical bonds, and electronic states of chromophores and interactions between molecules, etc. 5. Interactions between the substrate and the first layer. Infrared spectroscopy is useful not only for structural characterization of an LB film, but also usable to assess whether the LB film has a high quality based on the knowledge (1) to (3).

8.6.2 What we can learn from infrared spectra of LB films We will describe in more detail what we can learn from infrared spectra of LB films, citing an example of an LB film of 2-octadecyl-7,7,8,8tetracyansquinodimethane (octadecyl-TCNQ) (see Fig. 8.18). Figure 8.19 shows infrared spectra of octadecyl-TCNQ in a powder, in a bromoform solution, and in a ten-layer LB film (the Y-film) [107]. In general, when we measure an infrared spectrum of dye molecule with a long hydrocarbon chain such as octadecyl-TCNQ, we typically observe infrared bands due to the hydrocarbon chain and bands due to the chromophore. As the former, we can expect bands due to CH3 degenerate stretching, CH3 symmetric stretching, CH 2 antisymmetric stretching, CH 2 symmetric stretching, CH 2 scissoring, and CH2 rocking vibrations. In the bottom spectrum of Fig. 8.19, bands at 2955, 2918, 2847, 1462 and 1417 cm-l are assigned to CH3 degenerate stretching, CH2 antisymmetric stretching, CH 2 symmetric stretching and CH 2 scissoring vibrations (CH 2 scissoring vibrations appear as a doublet). Although a band due to CH3 symmetric stretching vibrations should appear in the vicinity of 2875 cm-l, this band is weak and therefore cannot be recognized in the spectrum [107]. Furthermore, a band arising from the CH 2 rocking vibration is generally expected to appear in the vicinity of 725 cml. Infrared bands due to the chromophore portion can be classified into bands assigned to in-plane and out-of-plane vibrations. Bands at 2223, 1546, and 1530 cm-' in the spectrum of the LB film are all bands due to in-plane vibrations of TCNQ portion, and assigned to C-N stretching, C=C stretching and C=C stretching vibrations, respectively [107]. In general, for analysis of an infrared spectrum of an LB film, we usually identify bands due to a hydrocarbon chain first, and thereafter look for bands arising from a chromophore. However, it is sometimes not easy to distinguish a band due to CH2 scissoring vibrations from a band due to the chromophore. In such a case, we may be able to distinguish 239

1

|

h

x

W ILI z

a: co m 0

C,)

4

t1 4000

3600 3200

2B00

2400

2000

1600

1200

WAVENUMBER(cm') Fig. 8.19. Infrared transmission spectra of octadecyl-TCNQ in a powdered microcrystalline state (top), octadecyl-TCNQ in a bromoform solution (middle), and 10-layer LB film of octadecyl-TCNQ deposited on both sides of a CaF 2 substrate (bottom). (Reproduced from Ref. [1071 with permission. Copyright (1991) American Chemical Society.)

them by measuring a spectrum of chromophore only which does not have a hydrocarbon chain. Now, what kind of information does a spectrum such as that shown in the bottom of Fig. 8.19 provide? First, we can obtain information regarding the conformation of a hydrocarbon chain from the frequencies of CH 2 antisymmetric and symmetric stretching bands [105,106]. These bands are known to appear in the vicinity of 2918 cm -1 and 2848 cm - l, respectively, when the hydrocarbon chain assumes trans-zigzag structure but shift to the higher-wavenumber side if the hydrocarbon chain contains some gauche forms. Hence, the result 240

shown in Fig. 8.19 tells us that the hydrocarbon chain of octadecyl TCNQ has trans-zigzag conformations while in the LB film and solid powder but contains a considerable number of gauche forms while in a solution [107]. We can learn about subcell packing of hydrocarbon chains from bands due to CH 2 scissoring mode [103,104]. The CH 2 scissoring vibrations appear as a doublet at 1471 and 1462 cm-l when the hydrocarbon chains take orthorhombic subcell packing, but as a single band at 1467 cm -l when the chains assume hexagonal subcell packing. Since bands due to a CH 2 scissoring vibration appear as a doublet in the top and bottom spectra of Fig. 8.19, it is considered that the hydrocarbon chains of octadecyl-TCNQ assume orthorhombic subcell packing both in the solid powder and the LB film (in the spectrum of the solution in Fig. 8.19, the CH 2 scissoring vibration appears as a singlet band as it is naturally expected). However, special care must be taken for the LB films of octadecyl TCNQ where the hydrocarbon chains assume interdigitated and non-interdigitated parts. Morita et al. [108] assigned the two bands at 1471 and 1462 cm-l of the LB films of octadecyl-TCNQ to the CH2 scissoring modes of non-interdigitated and interdigitated parts of the hydrocarbon chain. If one wishes to study the molecular orientation in an LB film, one must compare an infrared transmission spectrum with an infrared RA spectrum (Chapter 7.2.4). Let us introduce a simple example of comparison between a transmission spectrum and an RA spectrum. Figure 8.20 shows a transmission spectrum and an RA spectrum of seven-layer LB films of cadmium stearate [102]. We can readily notice the remarkable differences in the intensities of infrared bands between the two spectra. It is these differences in the intensities that allow us to discuss the molecular orientation in an LB film. Now, let us consider which bands will appear strongly in the transmission spectrum on an assumption that the molecular axis of cadmium stearate is nearly perpendicular to a substrate (see Fig. 8.21). In the case of a transmission method, since an electric vector of an infrared ray is parallel to the substrate, strong bands are those due to vibrations whose transition moments are perpendicular to the molecular axis, such as CH 2 antisymmetric and symmetric stretching vibrations (2919 and 2851 cm-l in Fig. 8.20, bottom), COO- antisymmetric stretching vibration (1543 cm l) and CH 2 scissoring vibrations (1473 and 1463 cm-l), whereas bands whose transition moments are parallel to the molecular axis, such as COO symmetric stretching vibration 241

v, COO-

0.02 vsCH2 vCH2

I Co

E

P V

c2

3-fi -

1 -8 R00 0 -

ll

I C00-l IcoCH2 r

I 0

-

_ _1800

_ _1200

__

I

Transmission vaCH2

0

co Co m) then a solution for b can be obtained by multiple linear regression (minimising the length of the residual vector E (/el )). The solution (Eq. (9.6)) is obtained by forming the generalised inverse of X. XTy =XrXb (X)-'X

T

y = (XTx)-XTX b

b = (XX)-X T y

(9.4) (9.5) (9.6)

We have to remember at this stage that only non-singular (the determinant is non-zero) square matrices have inverses. The technique shown above in multiple linear regression is to make the data matrix X into a square matrix (nxn matrix) by multiplying by its transpose X . However, the inverse of this product matrix can exist only when the resulting matrix is non-singular (or has full rank; see appendix). When the measurements are made in infrared spectroscopy, it can be seen that there is a lot of redundant information in the data. It means that there is 'collinearity'in the data profiles and the underlying variables (latent variables) responsible for the variation in the data are fewer than m. Modern multivariate techniques such as principal component analysis (PCA) and partial least squares calibration (PLS) are developed to handle data with such problems. In multivariate data techniques, the aim is to decompose the data matrix into the product of two matrices that can give us information on variables and objects. The criterion for the decomposition varies depending on the type of model one is trying to build with the data measured.

290

...

k01J

Fig. 9.2. Decomposition of the data matrix X into score matrix T and loading vector matrix P. The figure illustrates the data matrix containing five measurements reduced to the product of three score vectors and three loading vectors. Dimensionality of the matrices X, TPT and E are the same.

9.2.3 The data matrix X and its decomposition The data matrix X can be written as the sum of A matrices of rank 1 (Eq. (9.7)) and a residual matrix E. X=X 1 +X 2 +X3+ ... +XA+E

(9.7)

The decomposed matrices can be written as the products of score vector ta and a loading vector pa (Eq. (9.8)). X = tlpl'+t2P2' + t3'

... + tAPA' + E

(9.8)

Equation (9.8) is illustrated in Fig. 9.2. X = TPT +E

(9.9)

9.2.4 Graphic representation of spectral profiles The data matrix containing the spectral profiles of n objects and m variables can be viewed in two different ways (1) objects in variable space and (2) variables in object space. In the first case the objects (n) are represented as data points in M dimensional (m orthogonal dimensions) variable space. In the second case the variables (m) are represented as data points in n dimensional (n orthogonal dimensions) object space. Graphic representations of more than three dimensions are impossible on paper. A plot obtained for the objects in the variable space displays quantitatively the relationships between the objects, and the variables plotted in the object space displays the relationships 291

between the variables. All the information contained in the data matrix can be explained by these two plots. ·I Column vector (variable vector) Row vector (object vector)

->

X1 1

X1 2

X 13

X1 4

.

X2 1

X2 2

X23

X24

..

X2

X,3

Xn4

.

Xm

I

X2m .

X3 1

Xnl

.

....

. .

.

X3m

.

.

Xnm

9.3 PRINCIPAL COMPONENT ANALYSIS (PCA) Principal component analysis is one of the several multivariate techniques to decompose the total data matrix into data matrices containing information regarding the system (objects and variables) we are dealing with. The criterion used for the decomposition of the data matrix X is to extract latent variables in the direction of maximum variance in the data; that is, the first latent variable-the first principal component (PC 1)-is a linear combination of the original variables that explain the largest variance in the data X. The information explained by this latent variable is then removed from the data matrix. The second latent variable-the second principal component (PC2)-is then extracted in the direction that explains maximum variance in the rest of the data. These two principal components are orthogonal to each other. The decomposition of the data matrix X into principal components represents an ordinary least-squares solution which minimises the residuals E. One can extract several principal components to explain as much as possible variance in the data. The sum of the residuals goes down with the number of principal components extracted and reaches a minimum and then increases with the 292

extraction of more principal components (overfitting) [2]. The number of principal components needed to explain the information in the data set could be calculated by a process called cross-validation [2]. If the data matrix is column centred (i.e., each variable is adjusted in relation to the average of the variables), the origin of the variable space moves to the point represented by the averages of the variables in the data matrix. The principal components then pass through this point. We understand that the principal components are linear combinations of the M original variables. The projections of objects on the principal components (latent variables) are called scores. The scores of the objects on the principal components can be calculated by the scalar product between the unit vector (Wa) along the respective principal component and the object vector. ta =Xwa

(9.10)

All these scores are extracted in the score matrix T. The scores describe the similarities between the objects. These scores can be presented in the form of a projection plot on the plane containing PC1 and PC2, or PC2 and PC3, or PC1 and PC3. These plots are called score plots (Fig. 9.3). The samples that are similar group together in the score plots. The samples that are atypical to the other samples in the set will be isolated

X2 xlCos P

PC1

Fig. 9.3. Scores are obtained by projecting object vectors onto the principal components.

293

in the score plots and can be easily identified. These samples are called 'outliers'. The interpretation of a single latent variable and of the features of a latent variable model is possible through the connection to the original variables. This information is best displayed in loading plots. The coordinates of a variable in a loading plot are its loadings on the principal components. A loading plot displays the contribution of a variable to a model directly (proportional to the square of the distance from the origin). Variables that are lying near the origin contribute little and variables that are lying far away from the origin contribute more in the discrimination of the samples. Variables located in the same direction of a principal component carry similar information. Scores and loadings can be displayed simultaneously in plots called biplots [3]. These plots give information on the similarity between the samples and the variables that are responsible for the discrimination of the samples.

9.4 MULTIVARIATE CALIBRATION 9.4.1 Partial least-squares (PLS) calibration In many applications, one wants to relate some measured property y to a spectral profile x of intensities. The task is performed by measuring several pairs of(y, x) and subsequently constructing a model such that y=fAx)

(9.11)

The model is commonly calculated using least-squares procedures under the constraint of linear models. When there is only one property to be correlated with the spectral profile the PLS model building process is called PLS1, and PLS2 when there are more than two. In PLS1, the data is modelled in two stages, namely compression and calibration. In the data compression stage the spectral data are modelled in terms of a set of common latent regression factors {tl, t 2, t3 ... tA) (scores). These are n dimensional orthogonal column vectors for a data set with n samples and m spectral variables. These factors describe the major variations in the spectral data and at the same time are relevant for predicting the dependent variable. The compression and calibration steps can be written as follows for column centred spectral and response variables (see Fig. 9.4): 294

x·

I

UI

r

I

Fig. 9.4. A figure showing the decomposition of a data matrix X in principal component analysis and partial least-squares calibration. The details regarding the decomposition in these techniques are given in the text.

X= t

1

+t

2

+ t3p 3 ... '

+ tAPA' +E

y = tlql+tq 2+t 3 q3 ... + tAqA +F

(9.12) (9.13)

In the PLS1 algorithm, the vectors t, p and q are calculated starting with an equation relating the spectral data and dependent variable. X=yw' +E'

(9.14)

wl is a column vector (weightings) and is estimated as follows: w 1'=y'X/ y'X

(9.15)

w, is then normalised to unity. The estimated w, is then used to calculate t, Pl and q1 [4]. These estimates are again used to calculate the residuals of spectral and dependent variables. The process is repeated untilA factors which minimise the prediction error are found. The prediction of an unknown sample from the spectral profile xi proceeds as follows. First score t is found using the estimate for w1 and equation Xi = tl w' + el

(9.16)

Then the next score t2 is calculated from the solution of the above equation and the estimate for w2. The process is repeated until the Ath factor. The dependent variable is predicted by y = tq, + t2 q 2 + t3 q 3... + tAqA

(9.17) 295

Alternatively the same prediction can be written as y =Xb

(9.18)

In PLS calibration a generalized inverse X+ is constructed as b =X'y

(9.19)

and the calibration coefficients vector b is determined by Eq. (9.20) [5] b = W'(PW')-1 q

(9.20)

The parameters b are estimated so as to predict the property y as well as possible, for instance by means of cross validation [2]. By use of the parameters b the propertyy can be predicted from the infrared profiles. The predicted values of the property y for the training set samples is obtained by inserting Eq. (9.19) into Eq. (9.18): y =XXy

(9.21)

If there is a strong relationship between the property y and the intensities at some wavelengths, the predicted and the measured property y will be similar and the correlation coefficient between measured and predicted y will approach 1. 9.4.2 An application of multivariate calibration (PLS): determination of coal maturity (rank) Maturity or coal rank is used extensively by geochemists to characterise coal and kerogen samples [6]. Several methods have been used to determine maturity, including vitrinite reflectance, spore colour estimation, isotope ratios and chemical analysis such as biomarkers [7]. Coal consists mainly of three types of macerals: vitrinite, exinite, and inertinite. There is a correspondence between these three maceral groups and the chemical composition. Coal petrologists and chemists have linked the coal rank with maceral reflectance measurements [8]. Vitrinite reflectance is the most widely used method for maturity determinations. However, the analysis is time consuming and subjective [9]. Additionally, vitrinite is only one of the organic components in the sample and is not the major oil precursor. Assessing maturity by 296

measuring the major oil precursors or determining the chemical composition of geological specimens should provide more meaningful and accurate characterisation for petroleum geological purposes. Fourier-transform infrared spectroscopy was early taken into use for determination of maturity of kerogen [10-11] and coal [12]. The black or dark brown colour of the samples made it very difficult to analyse the samples by traditional transmission techniques. The diffuse reflectance technique then became popular for studying coal rank [13-14]. In these studies, specific functional group regions assigned to aromatic, aliphatic and C-H bending and stretching modes were correlated with chemical C/H ratios and rank. Fredericks et al. [15] used factor analysis to correlate specific parts of the FTIR spectra with various chemical and physical factors. In this example, randomly selected, vitrinite-rich coal samples from different parts of the world varying in vitrinite reflectance and in geological age were subjected to petrological, spectrometric and multivariate analysis. Twenty-five vitrinite reflectance measurements were made on each of the collected samples in the usual manner and then averaged [8]. The standard deviations for a selection of low-ranked coals (vitrinite reflectance 0.38-1.08) ranged from 0.03 to 0.13 vitrinite reflectance units, with an average of 0.06. Diffuse reflectance spectra of the samples were recorded in the range 4000-600 cm-l using 64 scans and a resolution of 4 cm-l. Three different spectra for each coal sample were averaged to give one spectrum for each sample. Each spectrum, consisting of 3401 data points, was transformed into Kubelka-Munk format [16]. Typical FT-IR spectra are given in Fig. 9.5 for five coal samples having different rank and vitrinite reflectance. It can be seen that the aliphatic/aromatic ratio, as determined by the ratio of the peaks at 2950 and 3050 cm l, decreases with increasing rank. It is difficult to draw qualitative conclusions about the other peaks, probably because of the interference from minerals. The wavenumber variables were reduced by approximately a factor of 10 through a maximum-entropy data reduction process [17-18]. The reduction process provides smaller matrices for computers to handle with concomitant decrease in the computing memory and time required and an increase in speed. After rejecting the abnormal and outlier calibration samples a cross-validated calibration model was established between the spectral 297

4000

3000

2000

1600

1200

600

Wavenumber cm' Fig. 9.5. Typical FT-IR spectra of coal samples in Kubelka-Munk format.

profiles and the vitrinite reflectance values. The total prediction error was then computed at the end and the number of PLS components giving the minimum prediction error was determined. Forty-six coal samples with vitrinite reflectances ranging from 0.38 to 1.08 were analysed. A total of five of these 46 samples were rejected, leaving 41 coal samples for the calibration. Multivariate calibration on the 41 samples gave an absolute prediction error of ±0.09 for determining the vitrinite reflectance of this 298

C t

i

0

1

2

3

4

5

6

7

8

9

Number of PLS components Fig 9.6. The progress of prediction error with the number of principal components extracted.

population. This prediction error is of the same order of magnitude as the average standard deviation of the corresponding vitrinite reflectance measurements (s = 0.06). This low prediction error is probably the best that can be obtained unless the measurement uncertainty in the vitrinite reflectance can be lowered. The model developed by cross-validation gave an optimum prediction using seven PLS components as evident by the minimum in Fig. 9.6, a plot of the Standard Error of Prediction (SEP) [2], vs. the number of PLS components. A plot of predicted vitrinite reflectance vs. measured vitrinite reflectance (Fig. 9.7) confirms the good performance of the model. The above results, obtained with coal samples from different parts of the world, produce one model that is generally applicable to vitriniterich coal samples with maturity within the range defined as the oil window (0.5-1.2). Thus, with the developed model and the combination of diffuse reflectance FTIR spectroscopy and PLS, we can predict the coal rank over the important oil production window on coal samples from different locations. The values of rank obtained are equivalent to those determined by vitrinite reflectance measurements, and are obtained with more objectivity, greater efficiency and less sample preparation. 299

I

·

/

/

I

I

( O

0

1.0 n

Cd Q e)

Q

0

U

0

Q

C 0.8 ,) o. a

0

0

Cd

Q2)

0 : 0s -

0

00o0

000 0

00 0

0. 6

'6 ._

0

O 0.4

0. 4

l I

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.1

Measured vitrinite reflectance Fig. 9.7. A plot showing the correlation between the measured and predicted vitrinite reflectance of the coal samples used in the analysis.

9.5 TARGET PROJECTIONS In certain applications, it will be of interest to find out which wavenumbers in the infrared profiles are most closely correlated to the property y. This can be done directly by calculating the covariance between measured y and the intensities at each wavenumber, i.e. by calculating the covariances ry,x as ry,, =ytX

(9.22)

For spectral profiles the covariances can be plotted as a covariance graph [19]. However, a better approach is to calculate the covariance between the predicted values of the property y and the intensities at each wavenumber: ry, = y tX

(9.23)

The result of the so-called target projection can be plotted as a graph to show the connection between a spectral profile andy. The point of using predicted instead of measured values of the property y in the correlation, is that wavenumbers of the infrared profiles with both high 300

predictive ability and high correlation with y are given increased importance compared to those that has only a good correlation. This can be shown by inserting Eq. (9.18) in Eq. (9.23): ry, = bTXTX

(9.24)

The product XTX is the variance-covariance matrix for the infrared profiles. Equation (9.24) can be expanded to provide m

ry,i

n

m

bjExkikj =bjrxixj; i -,2,...,m j=1

k=l

j=l

From Eq. (9.25), it is clear that the covariation ryxi between the predicted values ofy and the intensities at a wavelength y is calculated as a weighted sum of covariances ri,xj between intensities at two wavelengths i andj. The regression coefficients bj are used as weights. This shows that the target projection procedure weights the importance of prediction (bj) and correlation (rxi,xj) so that wavelengths that are important for predicting the property y have large variance (sensitivity) and are well correlated with other predictive wavelengths will be highlighted in the target projection plots. This is exactly the wavelengths that are important for the interpretation of the structural descriptor in relation to the property y [19]. Equation (9.25) further shows that for wavelengths where the covariance ry,i = 0, the variance in the intensities are probably zeros since it is quite improbable that a linear dependence should exist to make this correlation exactly zero. Thus, target projection can be used to find wavenumbers that gave the same intensity independent of the property we are examining. The target projection plots are easy to interpret and can be used to name factors influencing a system. Furthermore, for systems where variation in the multivariate data with a given dependent variable is small, target projection can amplify the changes in the target projection plots. In infrared spectroscopic data, the profiles are generally broad and overlapping. In many complex chemical systems the infrared spectra are featureless and contain very few broad bands. The changes in the spectra with external dependent factors are small and sometimes undetectable by visual inspection. Target projection analysis is very beneficial in such systems for interpretational purposes. 301

9.5.1 Understanding the dehydration process and band assignment of the overtone vibration of the water of crystallization of calcium oxalate monohydrate Calcium oxalate was among the first compounds to be studied by thermogravimetric analysis and it has been used as a standard for thermal analysis [20,21]. The diffuse reflectance infrared spectra of calcium oxalate monohydrate is shown in Fig. 9.8. The bands representing the H-O-H stretchings of the crystal water in calcium oxalate is broad over the range 3700-2600 cm - '. It exhibits five bands in the characteristic O-H stretching region. Band assignments of these peaks made by Petrov and Soptrajanov [22] are given in Table 9.1. Calcium oxalate crystal structure contains two non-equivalent oxalate ions and provides two different environments for the water molecules in the crystal structure. Furthermore, each water molecule has one of their OH groups involved in much stronger hydrogen bonding than the other. This should give rise to four O-H stretching bands. L

-

I -

o

(D I

I

Z z -o _ Li w R

I 4~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Mo 1,-)

O

r

I

I

oo00

3820

1

3240

2860

2480 2iOO NVENUMBE

;720

i

i

i340

_n

I

960

580

Fig. 9.8. A diffuse reflectance spectrum of calcium oxalate monohydrate in KBr (2% w/w). (Reproduced from Ref. [69] with permission.) 302

TABLE 9.1 Infrared band assignments of the OH stretching vibrations of the water molecules in the calcium oxalate monohydrate crystals CaC 2O4 .H2 0 Absorption (cmn1 )

Assignment

3486

v(OH) (2)

3428

v(OH) (1)

3336

v(OH) (1)

3250

26(OH) (?)

3058

v(OH) (2)

The fifth band, the lowest in intensity appears around 3258 cm - l . This is due to the overtone of the HOH bending mode reinforced by Fermi resonance [22]. The first and the last in the group (3486 and 3058 cm -l ) are due to one type of water molecule (type 2, as denoted by Petrov and Soptrajanov) and the remaining (3428 and 3336 cm-1) are due to the other type of water molecules (type 1). The oxalate stretching vibration bands appear around 1627, 1320 cm-l and bending vibrations band appear around 782 cm-l . Several authors have investigated the dehydration mechanism of calcium oxalate monohydrate [20-21,23-24]. However, none of them were able to demonstrate that there are two water molecular environments present in the crystal and their order of elimination during heating. In this application we will show that all these are possible in combination with target projection analysis. A Nicolet 800 FT-IR spectrophotometer and a diffuse reflectance accessory manufactured by Spectra-Tech, USA, were used for the spectral measurements. A high temperature-high pressure chamber (also from Spectra-Tech) was placed in the diffuse reflectance accessory in place of the ordinary sample cup. Calcium oxalate sample prepared as a 2% w/w in finely ground KBr was placed in the sample cup and heated to 80°C and held isothermally for 30 min to eliminate physically absorbed water from calcium oxalate and KBr. Then it was heated at a rate 5°C/min. and spectra were scanned at regular intervals. Each sample spectrum measured during heating was ratioed with the corresponding background (KBr measured under identical conditions) spectrum and the resulting relative reflectance spectrum [25] was transformed into Kubelka-Munk format. The area under the OH 303

*10-s

a

.a aD .W

l

Wavenumber cm-'

Fig. 9.9. The infrared spectral profiles of the OH stretching vibrations of 2% (w/w) calcium oxalate monohydrate in KBr measured at different temperatures. (Reproduced from Ref. [691 with permission.)

stretching bands was integrated. The area under OH stretchings was tested for linearity in advance and it can be used as a measure of the water molecules in the crystal. The diffuse reflectance infrared spectra acquired in the temperature range 80-155°C, the dehydrationprofiles obtained using the integrated area under the OH stretching peaks and the first derivatives of the dehydration profiles of calcium oxalate monohydrate sample (2% w/ w) at 5°C/min heating rate (rate of dehydration profiles after data smoothing) are shown in Figs. 9.9, 9.10 and 9.11, respectively. The derivative curves clearly show that there are at least two reactions taking place during the dehydration. At a minimum, there is a first water release step occurring between 90 and about 125°C and a second release step occurring between about 125 and 150°C. This observation is the first one of this kind. The shape of the reaction rate profiles suggests two different environments for the water molecules in the calcium oxalate monohydrate crystal structure, and may correspond to these different types of water molecules. 304

1.0 ·-

--.-

·

·--

·

'

i

0.8 e

--

W d

I

0.6

~/. I

I

=

0.4 0.2 N n

80

102.5

125

147.5

170

Temperature oC

Fig. 9.10. The dehydration profiles of the 2% calcium oxalate monohydrate sample in KBr. (Reproduced from Ref. [69] with permission.)

'1'.

0.04

a

/ I /

" 0.02

A A1

I

~

,

-

is_. .c

0.0 80

102.5

t

X

125

a! 147.5

_

_ 170

Temperature oC

Fig. 9.11. The rate of dehydration of 2% calcium oxalate monohydrate in KBr. (Reproduced from Ref. [69] with permission.)

In order to differentiate between the two different types of water molecules, we divided the dehydration rate profiles (derivative data) of 2% calcium oxalate sample (at 5C/min) into two subsets A and B (see Fig. 9.11). These two subsets indicate the temperature ranges of the dehydration where one can expect different types of reactions to take place. These temperature intervals in these two subsets were identified as 80-120 and 120-133°C. The raw spectral profiles of the water stretching vibrations in the temperature intervals 80-120 and 120-133°C 305

*10--

q

0.000

I

-0.080 -0.160 -0.240 2 -0.320 is -0.400 0.00 :

-0.07 -0.14 -0.21 -0.28 -0.35 3857

3517

3177

2837

2497

Wavenumber (cm-l) Fig. 9.12. Target projection plots obtained with dehydration spectral profiles in the temperature range: (a) 80-120°C; and (b) 120-133C. (Reproduced from Ref. [69] with permission.)

were calibrated against their respective temperatures and maximum correlating factors were obtained for each of the subsets by target projection [26-27]. These plots are shown in Fig. 9.12a and b, respectively. Target projection plots show the relative variation of each variable with temperature and each of the plots is an expression of the behaviour of the spectral data with increase in temperature. Zero line in the target plots separate the spectral profiles that is correlating positively and negatively with increase in temperature. The spectral profiles shown in Fig. 9.12 are below the zero line. This indicates that the peaks decrease in intensity (due to loss of water stretching vibrations) with increasing temperature, which is also obvious from Fig. 9.9. However, if the water molecules disappear at the same time and at the same rate then the target projection plots will be exact mirror images of the original spectral profiles. The figures show that this is not the case. Figure 9.12a shows without any doubt that the peaks at 3428 and 3336 cm -1 , which are due to type 1 water molecules, disappear at a faster rate than the type 2 water molecules (compare the depletion profiles of the 306

peaks at 3486 and 3428 cm-'). The plot in Fig. 9.12b shows the disappearance of the type 2 water molecules at a slightly faster rate than the type 1 (observe that the peak at 3486 cm-l has a maximum depletion profile). However, a higher rate of depletion of the peak at 3058 cm ' together with the peaks at 3428 and 3336 cm-l was not expected. This may be an indication that there is a peak relating to type 1 water molecules under the peak at 3058 cm-1 . They may be having overlapping maximums and seen as one peak in the infrared spectrum. Furthermore, these plots clearly show that the dehydration of both types of water molecules takes place at the same time but with different rates. Obviously, the type 1 water molecules are attached to the crystal structure with weaker hydrogen bonds than the type 2 water molecules. This is in agreement with the crystal structure determination of calcium oxalate monohydrate by Cocco et al. [28-29]. Petrov and Soptrajanov [22] in their analysis of the infrared spectrum of calcium oxalate monohydrate assigned the weak band (peak at 3258 cm-l) for an overtone of the bending vibrations of one of the types of water molecules. They were unable to assign the band to any specific type of water molecules because of the strong oxalate stretchings appearing in the same region as the bending modes of water molecules. Our target projection plots indicate that this band also decreases in intensity with temperature. This is reasonable because this overtone arises from one type of water molecules. A close analysis of Fig. 9.12 reveals that this overtone has a slow rate of disappearance during the first part of the dehydration (Fig. 9.12a) and a higher rate during the second part of the dehydration. This leads us to confirm that the overtone arises from the type 2 water molecules. The approximate maximas obtained with the dehydration rate profiles indicate that these occur around 0.3 and 0.75 conversion. These show that the two different types of water molecules are equimolar.

9.6 DATA ANALYSIS AND RESOLUTION BY ALTERNATING LEAST SQUARES If N spectra were obtained with mixtures containing A chemical components at M wavenumbers, they would define a two-way data matrix X of size N by M. If the components do not interfere with each other, one can express each spectrum as linear combinations of the spectra of the contributing components (Eq. (9.26)). 307

A

X =CS T +E =Zcsi

(9.26)

+E

i=1

where A is the number of chemical components. ST is the spectral matrix of the pure components (of dimension AxM) and C is the concentration matrix of dimension NxA. The experimental noise is expressed by the matrix E. The superscript, T, implies transposition of a column vector into a row vector. By applying Eq. (9.26), we assume that each measured spectrum adds up contributions from A pure species with concentrations defined by the concentration profiles ci, i = 1, 2, ..., A} and spectra by the spectral profiles {si, i = 1, 2, ..., A}. As mentioned above, the equation is valid for chemical components that do not react or interfere with each other. However, if there are interactions between different species, then one has to include a set of factors that model the non-linearity caused by the interactions. Then we can write Eq. (9.26) with additional factors that model the interactions. A

X =C *S

A'

+E =CisiT + E'Ci s'T +E i=1

(9.27)

i=1

The number of factors (A') needed to model the interactions, is dependent upon spectral similarity, similarity of spectral changes due to the interactions, as well as the noise level in the data set. Both additive and multiplicative errors like baseline shift and intensity variation due to, e.g., different path length for each spectrum, will contribute to the noise. The resolution of the component spectra and concentrations were accomplished by using the ALS [5,30] procedure. With the constraint of non-negative concentration and spectral profiles, the iterative process of calculating the least square estimate of the spectral profiles in S* is given by S *T = (C*TC*)- 1 C*TX

(9.28)

and the estimate of the concentration profiles in C* by C* =XS*(S*TS*) 308

1

(9.29)

For every cycle, negative intensities are set to zero. Prior to entering into a new cycle, the concentration profiles are normalised to sum to one as shown in Eq. (9.30). If one has any selective information regarding spectra or concentration profiles, the calculated values are substituted for by the selective information. The relative concentrations for each component can be calculated, taking into account that they should sum to one, by: A

Cb = cib

i

=1

(9.30)

i=1

and the constants necessary to scale the concentration profiles are found by least squares as b = (CTC)-iCT1

(9.31)

9.6.1 Resolution of infrared spectra and concentration profiles of the components of a multilayer laminate Polymers and plastics play a very important part in the life of 21st century man. A wide variety of things are made using plastics and polymers. Thin sheets containing multilayers of polymer components are used as packaging material in industry. The following example is to illustrate the analysis of such a polymer laminate using infrared microspectroscopy and alternatingleast squares. Application of chemometric techniques to the infrared microspectrometric data acquired from the laminate can reveal the spectra of individual layers and their concentration profiles. Furthermore, the chemical changes taking place at the interfacial regions can also be detected and their chemical information can be extracted in the form of the layer's infrared spectrum. The chemical changes taking place over a period of time can be monitored by comparing the infrared spectra of the layers at regular intervals. This will help the industry in determining the life span of the laminate. The problem at hand is a static multicomponent system because there are no dynamic changes in the concentrations of the components in the system when the measurements are made. The chemical composition varies across the cross-section of the laminate and this variation does not change during the analysis. 309

12x100o pm

apenure

/

r ..,.......--..---. .....1 ... ..1.11-1. I............... -... ....... --I. .... .1 ...... _-.1 ...... -.I.. . . .--. ...... ...... .11.

I laminate layers

1:x-.: .

Fig. 9.13. A sketch showing the redundant aperturing technique used in the analysis of polymer laminate.

I

.9

3500

3000

2500

2000

1500

1000

Wavenumber cm-' Fig. 9.14. A stack plot showing the infrared microspectroscopic spectra acquired by using the redundant aperturing technique. (Reproduced from Brune et al., Surface Characterization, 1997, with permission.) 310

The multilayer laminate sample was prepared by cutting a 5-pm thick cross-section using a microtome (Reichert-Jung, model 2050Leica). The sample was then mounted between NaCI windows in a compression cell (Spectra-Tech, Inc.). A small crystal of KBr was also placed in the same cell and this was used for collecting the background spectrum. The spectra of the laminate sample were collected at intervals of 2 lam, with a 12x100 pm2 sample area defined by redundant aperturing technique (Fig. 9.13). A total of 256 scans were co-added at a resolution of 8 cm-'. A total of 52 spectra of the laminate was collected in this way. The infrared microspectrometric data profiles of the 52 spectra (Fig. 9.14) were subjected to multiple component analysis using alternating least squares regression (ALS). The stack plot in Fig. 9.14 shows the presence of at least three components. The components arising from interactions and other underlying components are difficult to visualise in the data set. The analysis by alternating least squares resulted in five real components. The infrared spectra of the components are shown in Fig. 9.16 and their concentrations are given in Fig. 9.15. The components 1, 2, 4 and 5 are carbonated poly(vinyl chloride), poly(vinyl acetate), polyethylene and poly(vinyl dichloride) respectively. The component number 5 is an interaction product between poly(vinyl acetate) and polyethylene. The depth span of the components is 40, 20, 34 and 24 pm for the components 1, 2, 4 and 5, respectively. The interaction product (component 3) has a double distribution and spans about 60 pm.

Step-number

Fig. 9.15. The concentration profiles of the components resolved. The total concentration profiles across the laminate are normalised to unity.

311

Wavenumber, cm-' Fig. 9.16. The resolved infrared spectra of the components in the laminate sample.

312

9.6.2 Determination of the equilibrium constant and resolution of the HOD spectrum by alternating least squares and infrared analysis When water and deuterated water are mixed, a disproportionation reaction takes place and an equilibrium established with the product HOD as shown below H 2 0(1) + D20(1) = 2 HOD(1)

(9.32)

The equilibrium constant for the equilibrium is given by the equation

K = [HOD] 2/{ [H2 0] [D2 0] }

(9.33)

This equilibrium between water and deuteratedwater was first studied by Topley and Eyring [31] in 1934. The equilibrium in gaseous phase has been studied by several investigators [32-43]. The theoretical value of K for the equilibrium in gas phase has been determined from statistical mechanical calculations using measured and theoretical vibrational frequencies and anharmonicity constants obtained from infrared spectroscopy [34]. The best theoretical value (K = 3.85) for the equilibrium in the gaseous phase was determined by Wolfberg et al. [41] This led to a theoretical value of 3.88 for the liquid phase equilibria [44]. The interest in the theoretical determination of K was due to the difficulty in determining the equilibrium constant by experiment. This is because the species HOD can only exist in the presence of H 2 0 and DO, and the analysis of HOD in the presence of H20O and DO background is difficult. The use of infrared spectroscopy in the study of H 2 0 and D20O has been almost absent because of the strong absorptions of the OH/OD fundamental stretching vibrations. These vibrations generally give very broad absorptions due to extensive intermolecular bonding in the bulk of the liquid and created difficulty in obtaining the pure infrared spectrum of HOD by subtraction routines. When the modern sampling techniques such as total internal reflectance became available [45], spectra with reasonable intensities of absorptions that are suitable for quantitative analysis could be obtained. However, the resolution of the infrared spectrum of HOD required the use of the statistical equilibrium constant of 4 [46]. This 313

was needed to calculate the concentrations of the components in the mixture so that subtraction of H 2 0Oand D 2 0Ospectra could be made. In this application, the alternating least squares technique was used to resolve the infrared spectra of H 2 0O,D 2 0Oand HOD present in the equilibrium mixture, their concentrations and to determine the equilibrium constant K for the reaction. Furthermore, the resolved HOD spectra was used in assigning the bands. The change in the concentrations in the equilibrium mixture was achieved by changing the proportions of water and deuterated water. A macro circle cell manufactured by Spectra-Tech was modified in our laboratory to suit our experimental set up [47]. Samples were taken approximately to suit previously calculated amounts that could allow the analysis of the mixture in the range that is gradually changing from mole ratio 1 - 0 (for water) and 0 - 1 (for D 20 sample). The experiment started with a particular amount of water in the cell. The equilibrium concentrations were changed by adding deuterated water gradually in the cell. The D 2 0 sample was measured in the cell alone to achieve mole fraction 1 for D2O. A Nicolet 800 FT-IR spectrophotometer equipped with a medium band MCT detector was used to acquire the infrared spectra. A total of 100 scans were made each time in the range 4000-650 cm-l at a resolution of 1 cm- '. The spectra were then transformed into log(l/R). The infrared spectra of pure water, deuterated water and a mixture of water and deuterated water are shown in Fig. 9.17. The infrared spectra of the pure components and mixtures in log(l/R) format were transferred to a PC for processing and data handling. The spectral profiles were subjected to alternating least squares technique. With the assumption that the hydrogen bonded structures in water do not change upon isotopic dilution, it appeared that we needed only three components in order to describe the equilibrium between H2 O, HOD and D2O. Collecting the measured mixture spectra in a matrix X, the matrix can be expressed as a product of a concentration matrix C and a spectral matrix S, as shown in Eq. (9.26) (omitting the experimental error matrix, E). The dimensions of the matrices are as given in Eq. (9.26). The concentration profiles of the equilibrium mixtures can be obtained by Eq. (9.29). During the iterative procedure, for every cycle, negative intensities in the spectra were set to zero. In addition to the non-negativity constraints used by Maeder and Zuberbuehler [30] and Karjalainen 314

I

DO

WAVENUMBER, cm-

1

Fig. 9.17. The infrared spectra of pure water, deuterated water and a mixture of the two in log(l/R) format. (Reproduced from Ref. [31] with permission from Society for Applied Spectroscopy.)

[48] for the concentration profiles, we imposed the constraints of zero concentration of D 2 0Oand HOD when only H2 0Owas present and zero concentration of H2O and HOD when only D2O was present. Prior to entering into a new cycle, the concentration profiles were normalised. Red shifts of the peaks were observed during the addition of deuterated water to the water in the cell. This shift is due to the isotopic effects on peak positions upon dilution. This created problems in using only three components in the ALS procedure. In order to compensate for the shift an additional component was included in the iteration. The concentration profiles and the shift-factor were normalised to unit length before entering into a new cycle. The iterative process was terminated when the change in the difference between the measured and reconstructed mixture spectra was less than 0.01%. The first four score vectors obtained from a PCA-decomposition of the bending region (1800-1000 cm-l), were used as seeds for the concentration profiles and the shift-factor. The resolved concentration profiles, normalised to unit length, for H 2O, D2 0Oand HOD are plotted vs. mole fraction added H 2 0O,in Fig. 9.18a. Together with the concentration profiles the shift-factor's 315

z0

z 0

C)

MOLE FRACTION ADDED D2 0 Fig. 9.18. (a) The resolved concentration profiles together with the shift factor's variation with isotopic composition. (b) The molar concentration profiles of H2 0, D2 0 and HOD. (Reproduced from Ref. [31] with permission from Society for Applied Spectroscopy.) 316

i

6

-

5.5

5 ~~~~i

.

.

4.5

4

,

AI

3.5 3

IY

2.5

0.3

0.35

0.4

0.45

0.5

0.55

0.6

0.65

0.7

MOLE FRACTION ADDED D2 0 Fig. 9.19. The equilibrium constant K vs the molar fraction of added D2 0. (Reproduced from Ref. [31] with permission from Society for Applied Spectroscopy.)

variation with isotopic dilution is plotted. The molar concentrations of HO, D 2 0 and HOD are plotted in Fig. 9.18b. The molar concentrations were obtained from a least-square fit between the normalised concentration profiles and the known total concentration. The predicted molar concentrations were used to calculate the equilibrium constant according to Eq. (9.33). The average equilibrium constant was calculated to 3.86 + 0.07, based on the concentrations between 30 and 70% water. We avoided the values in the other regions in order to reduce the error in the equilibrium constant. A plot of K vs. molar fraction of added H 2 0 is given in Fig. 9.19. There is an excellent agreement both with values determined by experiment [39,40] and theory (K = 3.88) [44]. With the additional shift-factor incorporated in the iterative process, the resolved spectra of H2 0 and D2 0 fit almost perfectly with the measured spectra of pure H 2 0 and DO. The resolved spectra for H 2 0 and D2 0 correlates 99.99% with the measured spectra of the two pure analytes. 317

z

WAVENUMBER, cm-

Fig. 9.20. The resolved spectrum of HOD and the shift factor. (Reproduced from Ref. [31] with permission from Society for Applied Spectroscopy.)

The resolved HOD-spectrum is shown in Fig. 9.20 together with the spectral part of the shift-factor. There are three main bands appearing in the HOD-spectrum at approximately 3385 (VoH), 2490 (oD) and 1450 cm-l (v2). In addition there are two weaker bands at 2930 (2v 2) and 1850 cm-l (analogue to the association band in normal water (v2 + VL)). Isotopic dilution impose spectral shift in the bending regions of D 2 0O and H 2 0O.The shift is too extensive to be exactly portrayed by only one form factor and leads to the observed small derivative-like bands in the HOD spectrum in the vicinity of 1640 and 1210 cm 1. The resolved HOD spectrum has the same spectral features as the HOD spectrum resolved by Mar6chal [46].

9.7 TWO-DIMENSIONAL CORRELATION SPECTROSCOPY The direct observation of a correlation between an infrared band and a Raman band, the examination a correlation between a certain band and other bands in the same spectrum, etc. are now possible by generalized two-dimensional (2D) correlation spectroscopy, which we will describe in this section. In addition, this method allows one to highlight 318

various information which cannot be extracted easily from a ordinary one-dimensional spectrum. 9.7.1 Principle of two-dimensional correlation spectroscopy One may associate 2D NMR above anything else with 2D spectroscopy. It is true that 2D NMR is an essential tool today to analyze NMR spectra of complex compounds. However, 2D spectroscopy is not necessarily limited to NMR. In recent years, more and more people have started using 2D spectroscopy in various areas related to spectroscopy. Two-dimensional correlation optical spectroscopy, in particular, which was proposed by Noda [49-51] about ten years ago, is attracting increased attention as a new method for analyzing an infrared spectrum. The truth is that 2D correlation optical spectroscopy has an extremely wide variety of applications, ranging from various spectroscopic analysis including infrared spectroscopy to even x-ray diffraction. As shown in Fig. 9.21 [52], 2D correlation optical spectroscopy requires to expand a spectrum in both the X and Y axis directions (the

vvavenumDer, v1 Fig. 9.21. An example of a 2D correlation spectrum. 2D infrared-Raman heterospectral correlation map generated from temperature-dependent spectral variations of Nmethylacetamide in pure liquid. (Reproduced from Ref. [52] with permission. Copyright (1996) American Chemical Society.)

319

spectra to be drawn in the X and Y axis directions may be the same as each other or different from each other) and examines correlations between bands which appear in the expanded spectra. Studying the correlations, we can more clearly note spectral features (e.g., overlapping bands) which cannot be easily extracted from an onedimensional spectrum. Although the basic idea of 2D correlation optical spectroscopy is similar to that of 2D NMR, the methods of calculating correlation spectra are different [49-51]. In 2D correlation optical spectroscopy, a dynamic cross-correlation between intensity variations in bands induced by an external perturbation is calculated to thereby obtain a 2D correlation spectrum. We will explain 2D correlation spectroscopy in easier words with reference to Fig. 9.22 [51]. To obtain a 2D correlation spectrum, first of all, we must externally apply a certain perturbation (e.g., a time change, a temperature change, a concentration change) to our system of interest [51-55]. Subjected to the perturbation, components contained in the system generally respond differently from each other. To observe the responses, the system is irradiated with an electromagnetic wave. In other words, a series of spectra are measured. Now, assume that we applied a time change. In this case, we can obtain timedependent spectra. As we will explain using formulas, to obtain 2D correlation spectra, it is necessary to calculate dynamic spectra (Fig. 9.22). Based on the calculated dynamic spectra, we thereafter calculate 2D correlation spectra. While Fig. 9.22 shows a thermal change, a chemical change and various other changes as an external perturbation, 2D correlation spectroscopy, when initially proposed, could be applied only to an infrared signal which changes sinusoidally with time [49,50]. Although very effective for studying a system which is applied with a small external mechanical or electric perturbation as in the case of stretching a polymer film [56], the initial 2D correlation spectroscopy was sub-

£ Perturbation '

Electro-magnetic probe (eg, IR, UV) System

Mechanical, electrical, chemical, magnetic, t.. optical, thermal, etc.

Dynamic spectra

Fig. 9.22. A general scheme for constructing generalized 2D correlation spectra. (Reproduced from Ref. [51] with permission. Copyright (1993) Society for Applied Spectroscopy.)

320

jected to a restriction that a change of a dynamic spectral intensity with time (waveform) must be a simple sinusoidal wave. Hence, applications of the initial 2D correlation spectroscopy were rather limited. To remove the above restriction and further generalize 2D correlation spectroscopy, Noda [51] proposed generalized 2D correlation spectroscopy based on a new mathematical algorithm in 1993. The new 2D correlation spectroscopy is applicable to any waveforms, and hence, usable to various types of perturbations [51-55]. Further, the new calculation method is readily applicable to a variety of spectroscopic methods. In addition, generalized 2D correlation spectroscopy can be easily developed into hetero 2D correlation spectroscopy such as infrared-Raman and infrared-near infrared. 9.7.2 Synchronous and asynchronous correlation spectra We will describe the principles of generalized 2D correlation spectroscopy in more detail. Assume that we apply some perturbation which changes with time to a system. In this case, we obtain a series of spectra y(v, t) (where v denotes an infrared wavenumber, a Raman shift, etc.) which change during a time period from -T/2 to T/2. Meanwhile, dynamic spectra y(v, t)] are calculated by subtracting a reference spectrum y(v) from the series of spectra yt(v, t)} y(vt)y(v,t) = v)...-T...............T / ..................... and the other

(9.34)

Although the selection of the reference spectrum is somewhat arbitrary, in general, an average spectrum as follows is used Y(v)=

12

y(v,t)dt

(9.35)

In the case we use the formula (9.35), the dynamic spectra are deviations from the average of the spectra. Let us explain dynamic spectra by showing an actual example. Figure 9.23(A) shows temperaturedependent changes in near-infrared spectra of Nylon-12 [57]. In the region from 6000 to 5500 cm-l, we can find bands due to first overtones of CH2 stretching vibrations of the alkyl chain of Nylon-12. Since the conformation of the alkyl chain changes with temperature, the 321

A

B

wavenumber, v Nylon 12 (30*C - 150GC)

Fig. 9.23. (A) Temperature-dependent near-infrared spectra obtained from 30 to 150°C in the 6000-5500 cm- 1 region of Nylon-12. (B) Dynamic near-infrared spectra calculated from original near-infrared spectra shown in (A). (Reproduced from Ref. [57] with permission. Copyright (1997) American Chemical Society.)

intensities of the first overtones may change accordingly. This, however, is not very clear in Fig. 9.23A. As soon as we calculate the dynamic spectra (induced by the temperature changes in this example), we can find the spectra as shown in Fig. 9.23B and immediately tell which bands change [57]. 322

Now, in order to obtain generalized 2D correlation spectra, it is necessary to Fourier-transform the dynamic spectra measured in the time-domain into the frequency domain. The following is Fourier transform of dynamic spectral intensity changes y(v1 , t) observed at some spectral variable v1.

Y(Co) =

y(v,te itdt

= ylRe(o) + iIm

(9.36)

())

In the formula, yRe () and Y Im (io) denote a real part and an imaginary part, respectively, of the complex Fourier transform of y(v1, t). The Fourier frequency co represents the individual frequency component of the time-dependent variation of (vl, t). In a similar manner, Y2 *(co), the conjugate of the Fourier transform of dynamic spectral intensity, y(v 2, t), at spectral variable v2 is expressed as: Y2 (o) =| Y(v2,t e+itdt

(9.37)

= YIRe(o) -iYLm(o)

Once we find the Fourier transform, Y,(co) and Y* (co) of the dynamic spectra in the time domain measured at v1 and v2, respectively, we can calculate the complex 2D correlation intensity between them by the following formula. X(v

i(),v Y 2* ()don

(9.38)

Equation (9.38) is a formula for calculating a dynamic cross-correlation between spectral bands. The formula (9.38) consists of a real part 'D(vl, v2) and an imaginary part i(v 1 , v2) as shown in the formula (9.39): X(V1, V2 ) =( (v 1, V2 ) + i(V ,1 V2 )

(9.39)

The real part 4)(vl, v2) and the imaginary part i(v,, v2) are called the synchronous and asynchronous correlation spectra of the dynamic spectral intensity variations, respectively. In other words, these represent that time-dependent changes in the spectral intensities at the two frequencies v1, v2 are in-phase to each other (synchronous correlation 323

intensity) or out-of-phase to each other (asynchronous correlation intensity) [51]. While we consider time-dependent changes as perturbation in relation to the formulas (9.34) through (9.39), since classic time series analysis allows us to replace time with any other continuous variables, it is possible to calculate 2D correlation functions corresponding to various types of external stimuli such as a temperature change, a pH change and a pressure change instead of a time change. We will now explain the synchronous and asynchronous correlation intensities with reference to schematic diagrams. Figures 9.24a and b show 1D(v,, v2) and T(v,, v2) as two-dimensional contours which are called synchronous and asynchronous correlation spectra, respectively [51]. In the synchronous correlation spectrum, there appear on the diagonal line a few peaks called auto-correlation peaks which correspond to v = v2. The larger a band intensity change in response to an external perturbation (such as a temperature change), the stronger the intensity of an auto-correlation peak is. An auto-correlation peak always has the positive sign. Needless to say, a band having a strong intensity does not necessarily shows a strong auto-correlation peak. Hence, bands which overlap each other in a one-dimensional spectrum may be observed as separate bands in 2D correlation spectra because of different levels of responses to a perturbation. Peaks located at the off-diagonal positions of the synchronous spectrum are called cross peaks. The existence of a cross peak at (vl, v2 ) in the synchronous spectrum means that two bands at v, and v2 change in a similar manner to each other in response to a certain perturbation. A cross peak has the positive or the negative sign. A cross peak has the positive sign when the intensities of both bands increase or decrease with a perturbation. When one band increases while the other decreases, a cross peak shows the negative sign. An asynchronous correlation spectrum provides complementary information to information from a synchronous correlation spectrum. Of course, an asynchronous correlation spectrum does not show an auto-correlation peak. A cross peak at (v1, v2 ) in an asynchronous correlation spectrum means that the intensities of two bands at v, and v2 exhibit out-of-phase responses to a certain perturbation. For example, a cross peak appears when the intensities of two bands change at different temperatures. If an intensity change at v occurs at a higher temperature than an intensity change at v2, a cross peak located above the diagonal line has the negative sign. On the other 324

I ·0

c .5 M m U)

rn

Spectral variable, v,

r

o

Spectral variable, v

Fig. 9.24. (a) Synchronous and (b) asynchronous 2D correlation spectra constructed from dynamic spectra. One-dimensional reference spectrum is also provided at the top and side of the 2D map. (Original figures were prepared by Noda.)

hand, if the former occurs at a lower temperature than the latter, the sign of the cross peak is positive. This rule, however, is reversed if D(v, v2) < 0. One of the major features of generalized 2D correlation spectroscopy lies in asynchronous correlation spectra. This is because we can clarify in which order the intensities of various bands change if we analyze asynchronous correlation spectra. 325

9.7.3 What we can learn from 2D correlation spectroscopy The advantages of generalized 2D correlation spectroscopy are summarized as follows [51-68]. 1. Enhancement of apparent spectral resolution of overlapped bands. 2. Band assignments through observations of correlations between the bands. 3. Studies of inter and intra-molecular interactions through selective correlation between bands. 4. Probing the specific order in which the intensities of various bands change. Thus far, generalized 2D correlation spectroscopy has been applied, for example, to time-, temperature-, pressure-, concentration-, pH-, and phase angle-dependent spectral variations in the fields of infrared, Raman, near-infrared, visible, mass and fluorescence spectroscopy [51-681. Generalized 2D correlation spectroscopy has been utilized not only for basic research but also for applications such as those in biomedical sciences and food sciences. As an example of 2D correlation spectroscopy studies, we will describe a 2D infrared correlation spectroscopy study on the secondary structure of proteins using hydrogen-deuterium (H-D) exchange [58]. As described in Section 8.7.1, infrared spectroscopy has long been used as a valuable tool for qualitative and quantitative estimation of the secondary structure of proteins. Although the assignment of the components of the amide I band to secondary structure such as a-helix and 3-sheet has been the object of much effort, it is still a matter of controversy. Two-dimensional infrared correlation spectroscopy enhances the spectral resolution of the amide I and II regions and makes possible to assign some of the amide I and II bands to given conformations. Nabet and Pezolet [58] reported a 2D infrared correlation spectroscopy study of the secondary structure of myoglobin. They used H-D exchange of the amide protons as an external perturbation to generate the 2D synchronous and asynchronous spectra [58]. Because of the fact that the amide protons associated with each conformation are not exchanged simultaneously, the contributions from different conformations to the amide bands may be separated. The analysis of synchronous and asynchronous maps of myoglobin show that this method is very useful to unravel the different component bands under the poorly resolved amide I, II, and II' bands of proteins. 326

They prepared thin films of myoglobin of microgram quantity on an attenuated total reflection (ATR) crystal [58]. The H-D exchange was induced by hydrating the films with a flow of nitrogen containing D2 0O vapour. In general, there are two kinds of amide groups as to the kinetics of deuteration; some amide groups that are readily accessible to water are exchanged rapidly at the beginning of the deuteration process, whereas those involved in structures that are less accessible to the solvent show a slower exchange kinetics. Thus, in order to separate more efficiently the fast kinetics from the slower ones, different sampling time domains were used. Figure 9.25 shows A synchronous and B asynchronous 2D infrared correlation spectra of myoglobin calculated from the first 10 spectra recorded during the H-D exchange process [58]. In the amide I region of the synchronous correlation map for the rapidly exchanging protons (Fig. 9.25A) three correlation peaks appear at 1675, 1640, and 1615 cml. These amide I components are assigned to the -turns, random coil, and intermolecular P-sheets, often found in aggregated proteins, respectively. Therefore, it is very likely that the amide groups associated with these three conformations are exchanged first during the deuteration process. The strongest peak in the synchronous map is observed in the amide II region at 1530 cm-l, while in the amide II' region two major peaks can be identified at 1440 and 1350 cm-1 . The asynchronous map of myoglobin for the rapidly exchanging protons develops two cross peaks at 1675-1640 cm-l and 1640-1615 cm-l in the amide I region, confirming that the three peaks at 1675, 1640, and 1615 cm -1 appearing in the synchronous map are ascribed to three different conformations. Figure 9.26 depicts the corresponding synchronous spectrum calculated from 10 spectra obtained approximately 1 h after the beginning of the H-D exchange process [58]. It shows one autopeak at 1655 cm - , a frequency that is generally assigned to the amide I mode of the a-helix conformation. Thus, it seems that the amide protons of the a-helix conformation are exchanged more slowly than those associated with intermolecular -sheets, random coil, and 3-turns [58]. The other intense peaks observed at 1545 and 1345 cm- l may be assigned to the amide II and amide II' modes of the a-helices of myoglobin, respectively. The synchronous spectrum for the slow exchanging system (Fig. 9.26) also shows a weak component at 1625 cm-l. This component could be due to the 3-sheet conformation. Since the random coil and the turn structures do not develop the amide I components in the 327

50 1250

A

Wovenumbers, vt

B

1350

1450

1550

1650

1750

Woverurmbers,

Fig. 9.25. (A) Synchronous and (B) asynchronous 2D infrared correlation spectra of myoglobin calculated form the first 10 spectra recorded during the H-D exchange process. (Reproduced from Ref. [58] with permission. Copyright (1997) Society for Applied Spectroscopy.)

S Ike

3

E c3 C}

13 -A

Wovenurnbers,

Fig. 9.26. Synchronous 2D infrared correlation spectra of myoglobin calculated from 10 spectra measured approximately 1 h after the beginning of the H-D exchange process. (Reproduced from Ref. [58] with permission. Copyright (1997) Society for Applied Spectroscopy.) 328

synchronous map calculated for the long time domain, the H-D exchange rate for the P-sheet structure seems to be slower than those for the random coil and -turn structures.

ACKNOWLEDGEMENTS Some of the text and figures are reprinted from: D. Brune et al., Surface Characterization, 1997, pp. 410-424 (with permission from WileyVCH). F.O. Libnau and A.A. Christy, Determination of equilibrium constant and resolution of the HOD spectrum by alternating leastsquares and infrared analysis,Appl. Spectros., 49 (10) 1995 1431-1438; A.A. Christy, E. Nodland, O.M. Kvalheim, A. Burnham and B. Dahl, Determination of kinetic parameters for the dehydration of calcium oxalate mono hydrate by diffuse reflectance FT-IR spectroscopy. Appl. Spectros., 48 (5) (1994) 561-568 (with permission from Society for Applied Spectroscopy). REFERENCES 1. R.S. McDonald, Infrared spectrometry. Anal. Chem., 56 (1984) 349R-372R. 2. H. Martens and T. Naes, Multivariate Calibration. Wiley, Chichester, 1989. 3. K.R. Gabriel, Biometrica, 58 (1971) 953. 4. H.A. Martens, Multivariate calibration: Quantitative interpretation of non-selective chemical data, Dr. Techn. Thesis, Technical University of Norway, Trondheim, 1985. 5. S. Wold, A. Ruhe, H. Wold and W.J. Dunn III, SIAM J. Sci. Stat. Computat., 5 (1984) 735-743. 6. F.L. Staplin, W.G. Dow, C.W.D. Milner, D.I. O'Connor, S.A.J. Pocock, P. van Gijzel, D.H. Welte and M.A. Ytikler, How to Assess Maturation and Paleotemperatures, Short Course Number 7, SEPM, Box 4756, Tulsa, OK 74104, USA, pp. 1-289. 7. B.P. Tissot and D.H. Welte, PetroleumFormationand Occurrence-A New approachto Oil and Gas Exploration.Springer Verlag, Berlin, Heidelberg, New York, 1984. 8. E. Stach, M.Th. Mackowsky, M. Teichmiiller, G.H. Taylor, D. Chandra and R. Teichmuiller, Stach's Textbook of CoalPetrology, 3rd edn. Gebrtider Borntraeger, Berlin, 1982. 9. D.H. Dembicki, Jr., Geochim. Cosmochim. Acta, 48 (1984) 2641. 329

10. P.L. Robin and P.G. Rouxhet, Geochim. Cosmochim. Acta, 42 (1978) 1341. 11. T.V. Verheyen and R.B. Johns, Geochim. Cosmochim. Acta, 45 (1981) 1899. 12. T.V. Verheyen, R.B. Johns and R.J. Esdaile, Geochim. Cosmochim. Acta, 47 (1983) 1579. 13. M.P. Fuller, I.M. Hamadeh, P.R. Griffiths and D.E. Lowenhaupt, Fuel, 61 (1982) 529. 14. R.E. Anacreon, Application of diffuse reflectance for analysis of coal samples, Perkin-Elmer IR Bulletin, IRB-92, Instrument Division, Norwalk, CT, pp. 1-29. 15. P.M. Fredericks, J.B. Lee, P.R. Osborn and D.A.J. Swinkels, Appl. Spectrosc., 39 (1985) 303. 16. P. Kubelka and F. Munk, Ein Beitrag zur Optik der Farbanstriche. Zeitschriftfur Technische Physik, 11a, (1931) 593-601. 17. W.E. Full, R. Ehrlich and S.K. Kennedy, J. Sed. Petrol., 54 (1984) 117. 18. T.V. Karstang and R.J. Eastgate, Chemom. Intell. Lab. Syst., 2 (1989) 209. 19. O.M. Kvalheim and T.V. Karstang, Chemom. Intell. Lab. Syst., 7 (1989) 39. 20. R.L. White and J. Ai, Appl. Spectrosc., 46 (1992) 93. 21. L.A. Sanchez and M.Y. Keating, Appl. Spectrosc., 42 (1988) 1253. 22. I. Petrov and B. Soptrajanov, Spectrochim. Acta, PartA, 31 (1975) 57. 23. C. Bremard, C. Depecker, H. Des Grousilliers and P. Legrand, Appl. Spectrosc., 45 (1991) 1278. 24. E.S. Freeman and B. Carrol, J. Phys. Chem., 62 (1958) 394. 25. A.A. Christy, R.A. Velapoldi, T.V. Karstang, O.M. Kvalheim, E. Sletten and N. Telnaes, Chemom. Intell. Lab. Syst., 2 (1987) 199. 26. O.M. Kvalheim and T.V. Karstang, Chemom. Intell. Lab. Syst., 2 (1987) 235. 27. A.A. Christy, O.M. Kvalheim, F.O. Libnau, G. Aksnes and J. Toft, Vib. Spectrosc., 6 (1993) 1. 28. G. Cocco, Atti Acad. Naz. Lincei, Rend. Classe Sci. Fis. Mat. Nat., 31 (1961) 292. 29. G. Cocco and C. Sabelli, Atti Soc. Toscana Sci. Nat. PisaMem. PV Ser. A, 69 (1962) 247. 30. M. Maeder and A.D. Zuberbuehler, Anal. Chim. Acta, 181 (1986) 287. 31. B. Tropley and H. Eyring, J. Chem. Phys., 2 (1934) 217. 32. H.C. Urey, J. Chem. Soc., (1947) 562. 33. A.J. Narten, Chem. Phys., 41 (5) (1964) 1318. 34. J.W. Pyper and F.A. Long, J. Chem. Phys., 41 (1964) 2213. 35. A. Narten, J. Chem. Phys., 42 (1965) 814. 36. R.A. Weston, Jr., J. Chem. Phys., 42 (1965) 2635. 37. L. Friedman and V.J. Shiner, Jr., J. Chem. Phys., 44 (1966) 4639. 38. J.W. Pyper, R.S. Newbury and G.W. Barton, Jr., J. Chem. Phys., 46 (6) (1967) 2253. 39. A.J. Kresge and Y. Chang, J. Chem. Phys., 49 (1968) 1439. 330

40. V. Gold, Trans. FaradaySoc., 64 (1968) 2270. 41. M. Wolfsberg, A.A. Massa and J.W. Pyper, J. Chem. Phys., 53 (8) (1970) 3138. 42. J.W. Pyper and L.D. Christensen, J. Chem. Phys., 62 (7) (1975) 2596. 43. G. Jancso and W.A. Van Hook, Chem. Rev., 74 (1974) 689. 44. D.V. Fenby and A. Chand, Aust. J. Chem., 31 (1978) 214. 45. A. Lorber, Anal. Chem., 58 (1986) 1167. 46. Y. Marechal, J. Chem. Phys., 95 (8) (1991) 5565. 47. F.O. Libnau, A.A. Christy and O.M. Kvalheim, Appl. Spectrosc., 49 (10) (1995) 1431. 48. E.J. Karjalainen, Chemom. Intell. Lab. Syst., 7 (1989) 32. 49. I. Noda, Bull. Am. Phys. Soc., 31 (1987) 520. 50. I. Noda, Appl. Spectrosc., 44 (1990) 550. 51. I. Noda, Appl. Spectrosc., 47 (1993) 1329. 52. I. Noda, Y. Liu and Y. Ozaki, J. Phys. Chem., 100 (1996) 8674. 53. I. Noda, A.E. Dowrey, C. Marcott, Y. Ozaki and G.M. Story, Appl. Spectrosc., 54 (2000) 236A. 54. Y. Ozaki and I. Noda, Two-dimensional Correlation Spectroscopy. American Institute of Physics, New York, 2000. 55. Y. Ozaki, in: J.M. Chalmers and P.R. Griffiths (Eds.), Handbook of Vibrational Spectroscopy. Wiley, Chichester, 2001, in press. 56. I. Noda, A.E. Dowrey and C. Marcott, in: G. Zerbi (Ed.), Modern Polymer Spectroscopy. Wiley-VCH, Weinheim, 1999, pp. 1-32. 57. Y. Ozaki, Y. Liu and I. Noda, Macromolecules, 30 (1997) 2391. 58. A. Nabet and M. Pezolet, Appl. Spectrosc., 51 (1997) 166. 59. I. Noda, Y. Liu, Y. Ozaki and M.A. Czarnecki, J. Phys. Chem., 99 (1995) 3068. 60. S.J. Gadalleta, A. Gericke, A.L. Boskey and R. Mendelsohn, Biospectroscopy, 2 (1996) 353. 61. M. Muller, R. Buchet and U.P. Fringeli, J. Phys. Chem., 100 (1996) 10810. 62. I. Noda, Y. Liu and Y. Ozaki, J. Phys. Chem., 100 (1996) 8665. 63. K. Ataka and M. Osawa, Langmuir, 14, 951 (1998). 64. Y. Nagasaki, T. Yashihara and Y. Ozaki, J. Phys Chem., B, 104 (2000) 2846. 65. M.A. Czarnecki, P. Wu and H.W. Siesler, Chem. Phys. Lett., 283 (1998) 326. 66. C.P. Schultz, H. Fabian and H.H. Mantsch, Biospetrosc., 4 (1998) 19. 67. L. Smeller and K. Heremans, Vib. Spectrosc. 19 (1999) 375. 68. B. Czarnik-Matusewicz, K. Murayama, R. Tsenkova and Y. Ozaki, Appl. Spectrosc., 53 (1999) 1582. 69. A.A. Christy, E. Nodland, O.M. Kvalheim, A. Burnham and B. Dahl, Determination of kinetic parameters for the dehydration of calcium oxalate mono hydrate by diffuse reflectance FT-IR spectroscopy. Appl. Spectros., 48 (5) (1994) 561-568 331

Appendix I

Physical constants, conversion factors and atomic masses

A.

Some physical constants

Quantity

Symbol

Value

Units

Speed of light

c

Planck Constant

h

2.99792x103 6.62608xO134

ms - 1 Js

Avogadro constant

NA

6.02214x10 2 3

mo1-1

Atomic mass unit

u

1.66054x10- 2

kg

Electron mass

me

9.10939xl10-3

kg

Proton-mass

mp

1.67262x10 27

kg

Neutron mass

ms

1.67493x10

-2 7

kg

Elementary charge

e

1.60218x10-1

Gas constant

R

8.31451

7

9

C JK - 1 mol - 1

Conversion factors

B.

Quantity

Conversion factor

1 eV

1.60218x10-19J

1 cal

4.184 J

1 atm

101.325 kPa

1A

10-1° m -

1 Nm l

103 gl

333

C.

Some atomic masses

Element

Mass x10-2 7 kg

Hydrogen (H)

1.6738

Carbon (C)

19.9450

Nitrogen (N)

23.2587

Oxygen (0)

26.5676

Phosphor (P)

51.4332

Sulphur (S)

53.2369

Chlorine (Cl)

58.8867

Iodine (I)

210.7299

334

Appendix II

Some character tables and point groups

Cs

E

GxE

A'

1

1

x,yR z

xy,z,xy

A"

1

-1

zR,R,Ry

xzyz

Ci Ag

E 1

i 1

R,RyRz

x ,y z ,xy,xz,yz

A.

1

-1

Xy,z

C2 A

lE

B

C2(z)

i

1

ZR z

x2,y 2 z2,xy

1

-1

xyRx,y

xz,yz

C2v and C3v are found in the text of Chapter 4. C4,

E

2C4(z)

C2

2(,y

2Gd

A1

1

1

1

1

1

z

A2

1

1

1

-1

-1

Rz

B1

1

-1

1

1

-1

B2

1

-1

1

-1

1

E

2

0

-2

0

0

X2 +y2,z2

2

_y2

xy (xy) (Rx,Ry)

(yz,xz)

335

C5

E

2C 5(z)

2C 52

5y v

4=

41

1

1

1

1

z

A2

1

1

1

-1

Rz

E1

2

2cos4

2cos2

0

(xy) (R,Ry) (yz,xy)

E2

1

2cos2

2cos

0

x

C,,

72° 2

2

z ,X2 +y

2

y2,xy

E

2C,

...

ooC V

+ (A1 )

1

1

...

1

z

Z- (A2)

1

1

...

-1

Rz

n1(El)

2

2cos4

...

0

(x,y) (Rx,Ry) (yzxz)

A (E2)

2

2cos25

...

0

· (Eg)

2

2cos35

...

0

x2 +y2,z2

(x2 -y 2xy)

C2h

E

C2

i

ah

Ag

I

1

1

1

Rz

Bg

1

-1

1

-1

Rx,Ry

xy

Au

1

1

-1

-1

z

yz,xz

B,

1

-1

-1

1

D2

E

C2 (z)

C2 (y)

C2 (x)

A

1

1

1

1

Rz

z ,X ,y

B1

1

-1

1

-1

ZRz

xy

B2

1

1

-1

-1

y,Ry

xz

B3

1

-1

-1

1

x,Rx

yz

D3

E

2C3

3C2 '

Al

1

1

1

A2

1

1

-1

ZR

E

2

-1

0

(xy),(RxRx)

336

2

2,2,y

2

2

2

z2x 2 + y2 z

(x2-y2 ,xy),(yz,xz)

D4

E

2C4

C2

2C 2 '

2C2"

A1

1

1

1

1

1

A2

1

1

1

-1

-1

B1

1

-1

1

1

-1

B2

1

-1

1

-1

1

z,R x2_y 2 xy (xy),(R R,)

0

0

-2

z2,x2+y

(yz,xz)

E

2

0

D2 d

E

2S4 (z)

C2

2C 2 ' (x)

A1

1

1

1

1

1

A2

1

1

1

-1

-1

,B1

1

-1

1

1

-1

B2

1

-1

1

-1

1

z

xy

E

2

0

-2

0

0

(xy),(R,,R v)

(yz,xz)

D3d

E

2C 3

3C2 '

i

2S 6

3cd

Alg

1

1

1

1

1

1

A2g

1

1

-1

1

1

-1

Rz

2

-1

0

(Rz,Ry)

2

3d 2

z2,x2+y

R x2y

2

z2+y 2

Eg

2

-1

0

Alu

1

1

1

-1

-1

-1

A2 u

1

1

-1

-1

-1

1

z

xy

iE,

2

-1

0

-2

1

0

(x,y)

(yz,xz)

2_y 2

D2h

E

C2(z)

C2(y)

C2 (x)

i

oy

cz

y

Ag

1

1

1

1

1

1

1

1

Big

1

1

-1

-1

1

1

-1

-1

Rz

xy

B2g

1

-1

1

-1

1

-1

1

-1

Ry

xz

Bg

1

-1

-1

1

1

-1

-1

1

R

yz

AU

1

1

1

1

-1

-1

-1

-1

Bl1

1

1

-1

-1

-1

-1

1

1

z

B2.

1

-1

1

-1

-1

1

-1

1

y

B3 .

1

-1

-1

1

-1

1

1

-1

z

z2,x2y

2

337

D3 ,1

E

2C3

3C2'

Ch

2S 3

3c,

A1 '

1

1

1

1

1

1

A2

1

1

-1

1

1

-1

R,

E'

2

-1

0

2

-1

0

(x,y)

Al"

1

1

1

-1

-1

-1

A2 "

I

1

-1

-1

-1

1

z

E"

2

-1

0

-2

1

0

(Rx,RY)

D6h

E

2C6 1

lg A2g 1

2C 3 C2 1

1

1

1

3C 2' 3C 2 " i 1

1 1

1

2S

3

2S 6

3,

d

3V

1

1

1

1

-1

-1

Big

1

-1

1

-1

1

-1

1

-1

1

-1

1

-1

B2g

1

-1

1

-1

-1

1

1

-1

1

-1

-1

1

Ig

2

1

-1

-2

0

0

2

1

-1

2

0

0

2

-1

-1

2

0

0

2

-1

-1

2

0

0

1

1

1

1

-1

-1

-1

-1

-1

-1

-1

1

-1

1

-1

-11

-1

1

-1

1

2

1

-1

-2

0

0

-2 -1

1

2

0

0

2

-1

-1

2

0

0

-2

1

-2

0

0

Alu Blu

1

(x2-y2 ,xy)

(yz,xz)

=

2,X2+y2

1

11

1

11

,

z2

R (yz,xz) (R~,Ry) (x2-y2,xy)

lu

lu

D.h

E

2CO ...

ooa,

Z(Ag)

1

1

1

g(A 2g)

1

1 i 1

2S,

...

moC

1

1

1

-1

1

1

-1

[lg(Elg) 2

2cosl

0

2

-2cos

0

Ag(E2g)

2cos2?

0

2

2cos2o

0

1

-1

-1

-1

-1

1

-1

-1

-1

1

F[I(E1u) 2

2coso

0

-2

2coso

0

A(E 2 u) 2

2cos2o

0

-2

-2cos2o

0

2

,S(Alu) 1 XZ(A 2 u)

338

1

(x,y)

z2,x2+y2

R (RRy)

(yz,xz) (x2 -y2 ,xy)

(XY)

Appendix III

Matrices

A. A matrix A matrix is an array of elements of the following form. The horizontal sets of elements are called rows and the vertical sets of elements are called columns. A matrix is represented by a single symbol.

A = [aj] =

all

a,

2

a 13

all

a,.

a2 l

a2 2

a23

a2j

a2m

ail

ai 2

ai3

ai3

aim

ani

an 2

an

an

anm

The element aij is the entry belonging to the ith row andjth column of the matrix. The above matrix has n rows and m columns. These are called dimensions of the matrix. The above matrix is said to be an nxm matrix. When the dimensions are equal, the matrix will have an equal number of rows and columns (m = n) and the matrix is a square matrix with m 2 entries. Matrices follow certain rules and we shall learn more about their behaviour in the following sections.

B.

Matrix addition

Matrices of same dimensions can be added. IfA and B are two matrices of the same dimensions then the sum of the matrices is a matrix of the 339

same dimension as A and B. The resulting matrix C will contain entries cij which are given as cij = a + b. For example, the addition of matrices A and B are shown below

A= 0 1

-

1 2

B=

0

0

-1 2

1

2

0

-1

-1

0

A+B=

1 -1 1 3

3 -1

0

0

1

Matrix addition is commutative. It means that whether we add matrix B with A or A with B, the resulting matrix is the same. A+B=B+A=C

C.

Scalar product of a matrix

A matrix A can be multiplied by a real number k. The multiplication can be written as kA. The resulting matrix is of the same dimension as A and contains elements kaij as entries. For example 3A is

0

3

3A= 0 3 -3 3 6

D.

0

Matrix subtraction

The subtraction of a matrix B from A can be considered as an addition a A+ (-1)B. Matrices of the same dimensions can be subtracted from each 340

other. The resulting matrix is a matrix of the same dimension. The matrix would contain elements cij = aj - biv if B is subtracted from A or di = bj - aij ifA is subtracted from B. It is easy to see that the result of

the subtraction is not the same. For example, the subtractions would result into the following matrices. 1

1

--

2

-1 3

0

A-B=fi

E.

-1

-1

I

B -A = 1 -2 -3

0

Matrix multiplication

Two matrices C and D can be multiplied if the number of columns in the matrix C is the same as the number of rows in D. For example if C is an nxm matrix then D has to be an mxk matrix. The product of the matrices C and D is an nxk matrix. If the resulting matrix is E then the element ej is given by ei = Zcikdkj k=l

where m is the number of columns in the matrix C and number of rows in the matrix D. There is no product between two matrices that does not satisfy the requirement above. For example, if

C=['

2] and D =[

-1

1]

then the product E = CD is

CD =

5 2

For example, the product between matrices A and B given above is 341

0 1 0 -1 2 1 0 2 AB= 0 1 -1 1 2 0 = 2 3 12 0 -1 -1 0 2 3 2 The product between B and A is

0 -1 2 1 0 1 BA= 1 2 = 1 -1 -1 -1 0 1 2 0

2 3 1 1 2 -1 -1 -1 0

As we can see from above, the product of two matrices is not generally commutative. F.

Identity matrix

All square matrices have an identity matrix of the same dimension. The identity matrix has elements i, = 0 when i j and 6ij = 1 when i =j. The identity matrix is denoted by the symbol I. The identity matrix for all 3x3 matrices is then -1

0 O

I= 0 1 0 0

I

The product of a square matrix with its identity matrix is the square matrix itself. That is IA = A.

0

1 0

1

IA= 0 1 0 0 1 -1 0 0 1 1 2 0

1 0

1

0 1 -1=A 1 2 0

One can also show that the product of any square matrix multiplied by its identity matrix yields the same square matrix.

0 AI= 0

1

1

1 2 0 342

1 0 0

1 0

1 0 = 0

0 1

1

1 -1=A

1 2 0

G.

Transpose of a matrix

A matrix can be transposed so that the elements in the rows of the matrix become columns of a matrix. For example, the matrix D can be transformed into a matrix so that the elements dij become dji of the new matrix. The new matrix is the transpose and is denoted by D'.

]

D = [i -1 2

H.

3

D'= l-1 3]

1

Determinant of a matrix

Certain products of the elements compute determinant of a matrix. It can only be evaluated for a square matrix (number of rows = number of columns). For example the determinant of the matrix X=

xL

11

X 12

is computed as x11 x22

det(X)= XI = Xl l x2 1

-

x21 xl 2 and is denoted by

X12 22

where x22 and x12 are called co-factors of x1 and x21. When the number of elements increases, the evaluation of the determinant becomes tedious. For a 3x3 matrix

Z =

Z1 1

Z1 2

Z13

Z2 1

22

Z23

LZ31

Z32

Z33

the determinant can be evaluated by the sum of the products between the elements of a particular row or column and their co-factors. The determinant is then

343

det (Z) = z 11 Z 11 +

21

Z 21 + z31 Z31 = Zll (- 1)+l Z2 2 z32

2 (1)2+1

-

Z1 1 Z1 1 +

12

Z12

Z13

Z31 (_1)3+1 Z12

Z13

Z3 2

Z33

Z22

Z23

Z23 + Z33

Z1 2 + z1 3 Z13

The determinant of matrix A = 01

1

isthen det (A)+ 1(0-1) is then det (A)=A=11(0 + 2) -0 + 1(0-1)} = 1

I.

Cofactor

The co-factor of an element zij in a square matrix Z of dimension nxn is the determinant of the matrix of dimension [(n-1)x(n-1)] obtained by deleting the ith row andjth column from the matrix Z. The cofactor is then ()i+j I[zi]1. The co-factor is denoted by Zij. The co-factor matrix

J.

The co-factor matrix of a square matrix Z of dimension nxn is the matrix obtained with all the determinants of the co-factors of the elements in the matrix Z. The cofactor matrix is denoted by [Z,,]. The co-factor matrix of Z given above is then Z2 2

Z23

Z3 2

Z33

Z1 2 Z32

Z13 Z3

Z

Z13

Z2 2

Z23

Z2 1 31 Z1 1 Z31

Z2 3

Z2 1

Z2 2

Z33

Z3 1

Z3 2

Z1 3 33 33

Zll Z3 1

Z1 2 Z3 2

11

Z13

Zll

Z1 2

Z2 1

Z2 3

Z2 1

Z2 2

_Z

-

The co-factor matrix of the matrix given above is then 344

11

-1

0

20 0 1

1

-]2 0 01 11

1 21

2

-1

1

1 Il

1 2l = -2 11 0 -1

-1

-2

1

1

0 X

0 °1

I1 111;

o:2

-1

K.

0 1

-1

Inverse of a matrix

Inverse A -1 of a non-singular (non-zero determinant) square matrix A is a non-singular square matrix that satisfies the following

A-1A = I Inverse of a matrix A can be evaluated from its determinant and the transpose of the co-factor matrix. A- 1

I(1/det (A)][Aij]' 2 =

L.

2

-1

1 -1 -1 -2

Solution of simultaneous equations

The knowledge of matrices can be applied to solve simultaneous equations. For example consider the following set of equations. X1 + x3 = 3 X2 -X 3 = -3

x1 + 2x2 = 5 The above equations can be written as follows

345

X1 + OX2 + x 3 = 3 OX1 + x 2 - X3 = -3

x, + 2 x2 + OX3 = 5

In matrix form the equations can be written as

0 1

-1O

-3

x2=

The matrix at the right hand side is the same as matrix A given above. So the equations can be written as

Ax = y where x and y are x21 and -3, X3

respectively.

-1

The equations can be solved by multiplying the equation by the inverse of A. That is A-l Ax = A-l y Ix

=A-l y

x

=A-ly

That is, the solution for the above equations is x = A-ly

x2 = X3

-1

-1

1

-3 =-1

-2

1

-1

2

that is, x1 = 1, x2 = -1 and x, = 2. 346

M.

Eigen values

If P is a square matrix of order n, then the k matrix [P - IX] is called the characteristic matrix of P. The determinant IP - /I1 is called the characteristicdeterminant of P. The expansion of the above determinant expressed as a polynomial of degree n in X is called the characteristic function of P. The equation f(X) = 0 is called the characteristic equation of P and its roots are called the characteristicroots or eigen values. For example, for the matrix

[-2 2]' the determinant becomes 1-k

1

-2

2-X}

This leads to the equation (1 - X)(2 - X) + 2 = 0. The solution to the above equation gives values These are the eigen values of the matrix P.

N.

= 3 and X = 1.

Eigen vectors

For any square matrix P of order n, the eigen value equation can be written as Px = Xx where x is a lxn matrix and the X one of the eigen values. There are n solutions to this equation corresponding to n eigen values. This vector x is called an eigen vector of the matrix P. The eigen vectors can be found by solving the above equation.

O.

Trace (character) of a matrix

If A is a square matrix, the trace of A is defined as the sum of the elements on the principal diagonal. The trace is denoted by tr(A). The trace of matrix A given above is 347

tr 0 1 -1=2 112

P.

-_

Similarity transformation of a matrix

If Q is a non-singular square matrix then the product RQR - = F is known as a similarity transformationof Q by R. The matrices Q and R are similar. Their determinants, eigen values and traces are equal. tr(RQR 1 ) = tr(F) For example, if Q is 0

-1 2

1

2

(the same as B above) and R is

-1 -1 0

O1

-1 (the same as A above), then

1 2

0

RQR - = A BA-

1 0 1 1 -1 1= 1 2 0

0 -1 2 2 2 -1 -2 -4 1 1 2 0 -1 -1 1 = 1 11 -1 -1 -1 -2 1 -1 -33

det(ABA - l) = det(B) = 2 tr(ABA -1 ) = tr(B) = 2 eigen value equations ofABA - and B are the same as ;3 - 22 + 3R -2 = 0. 348

Q.

Diagonalisation of matrices

A non-singular square matrix Q can be diagonalised by another matrix R by similarity transformation (RQR - 1 = G) such that the resulting matrix G has only non-zero diagonal elements. These diagonal elements are the same as the eigen values of the matrix. The columns of the matrix Q are the same as the eigen vectors corresponding to the eigen values taken in the same order. Diagonalisation by similarity transformation is used in reducing the matrices representing operations of a point group. To illustrate this, let us use the matrices A, B, C, and D used in Section 4.6. The matrices A and C are in the diagonal form. The matrices B and D are the same. They have eigen values X = -1, -1 and 1. When the matrices are diagonalised they have -1, -1 and 1 as diagonal elements. The diagonalisation of the matrix B and D can be done by finding the eigen vectors of the matrices B and D. These matrices have one diagonal element (-1) in the middle row each. Therefore it is enough to diagonalise the first and the last rows

B=D=

-1

0

0

0

-1 For k = -1

[0 -- 1

[

01[X

The1 normalizing condition leads = to

2

+

The normalising condition leads to x2 + 2x 2 = 1

x = +1I

1

and z = 1/2.

For k = 1

349

l0x oo z]

0°1 -ol=-,Lo

-1 -iL

with normalising condition as above, we get x = +1/F2 and z = -1/42. The matrix Q that diagonalises the matrices B and D can be written as

o

Q/ 0 Q1/

-1

-1/2] 0

o

1/ j

-1/

QBQ -

=

2

0

0

o 1/2]and Q =1

-1/

0 11 =

-1

-1

22

0O o -1- 1/ O O -1 O

I-1

0

O

0

o 1/J2

Q l= 2

-1

L1/

0

/

0

-1/-

-1

0

0

1/]

0

0 0 -1 Therefore, the matrices A, B, C and D representing the symmetry operations E, C2, %a,and %z are reduced to

i

0O 0 0 0 1 0 0 1 0 O, 1 0 1 0,O -1 -1 0 , respectively. 1 0 and i OO 1O 0 0 1 0 0 -1

350

Index

absorbance, 21, 133, 285 absorption spectrum, 18 absorptivity, 131 accidental degeneracy, 95 acrylonitrile- butadiene-styrene (ABS), 165 alkyl chains, 252 allowed transition, 19 alternating least squares, 309 amide I, 142,255 amide II, 142,256 amplitude, 3 anharmonic constant, 35 anharmonic terms, 35 anharmonicity, 33, 37,38 anionic surfactant sodium dodecyl sulphate, 198 anisotropic distribution, 204 anti-Stokes lines, 79 anti-symmetric stretching vibrations, 29 anti-tumour agen, 170 apodization, 121 arsenic-doped silicon, 234 ascorbic palmitate, 261 assembled cells, 134 asymmetric stretching, 239 asynchronous correlation spectra, 321,325 attenuated total reflection (ATR) method, 131, 136, 141, 196

auto-correlation peak, 324 azobenzene, 220 bacterial cells, 263 bacteriorhodopsin, 255, 268 barbituric acid, 249 beamsplitters, 107, 116 bending vibrations, 29 benzaldehyde, 38 biological materials, 195 biomarkers, 296 biomembranes, 260 biplots, 294 bisacrylates, 215 bolometric effect, 110 bone disease, 227 breast tumour, 232 cadmium stearate, 241, 242 caffeine, 187 calcium oxalate, 302 carbon nanoparticles, 171 cassegrain type, 188 cellulose ether, 197 centreburst point, 120 cervical cancer, diagnosis of, 273 chalcogenide glass, 235 character table, 71 characteristic absorption bands, 31 charge-transfer reaction, 247 chemometrics, 256 351

chlorophylls, infrared spectra of, 259 chromophores, 238 chromophoric groups, 255 classical least-squares, 181 coal rank, 296 CO-ethylene- propylene alternating copolymers, 197 colon carcinoma, 233 combination modes, 33, 37 commutative, 59 concentration, 285 conducting polymers, 218 conjugate operations, 65 Connes advantage, 120 continuous-scan FT-IR, 124 continuum model, 151 corpuscular, 6 cross peaks, 324 crystallinity, 212 curve-fitting, 257 cycle, 3 cysteine residues, 255 degenerate stretching, 239 degenerate vibrations, 29 dehydration profiles, 304 dependent variables, 288 detectivity, 113 detector, 109 detector responsivity, 113 deuterated triglycine sulphate (DTGS), 106 deuterated water, 313 deuteration studies, 202 di(carboxystyryl)benzene, 219 diatomic molecule, 23 dichroic ratio, 204 diffuse reflectance (DR) method, 131, 147, 285 diffuse-reflected light, 148 diffusion depth, 156 dioctyl phthalate, 185 dipole moment, 82 352

direct deposition, 185 dispersive instruments, 117 dispersive spectrometers, 207 distinct operations, 45 dynamic FT-IR procedures, 207 dynamic IR spectroscopy of polymers, 212 dynamic spectral intensity, 321 elastomer sealing rings, 165 electric dipole moment, 20 electroluminescence, 219 electromagnetic spectrum, 1 emission spectroscopy, 131 epitaxial layers, 146 essential oil constituents, 183 etendue advantage, 120 exinite, 296 external perturbation, 320 fast Fourier transform, 127 fatty acids, 260 Fellgett advantage, 120 Fermi resonance, 38,95 fingerprint region, 18 fixed cells, 134 fluoranthene, 185 focal-plane arrays, 222 forbidden transition, 19 force constant, 25 Forman algorithm, 123 Fourier transform, 323 Fourier transform dynamic infrared spectroscopy, 208 frequency, 25 frequency domain, 125 FT-IR instrumentation, history of, 105 functional group, 22,285 fundamental vibrations, 17,80 fundamentals, 32 gauche conformation, 248

gauche forms, 240 GC/FTIR, 180 Globar source, 107 Golay detector, 110 ground vibrational state, 21 group frequencies, 18, 31, 97 Halobacteriumsalinarium,268 harmonic oscillator approximation, 23 harmonic oscillator model, 33 harmonic vibrations, 23 heavy water, 135 hemes, 255 Herman orientation function, 205 Hermite polynomial, 28 hexaalkoxytriphenylenes, 215 homomorphous, 59, 64 human melanoma, 233 hydrocarbon chains, 238 hydrogen bonds, 199 Hz, 3 identical operation, 43 improper rotation axis, 47 impulse-response technique, 207 incident light, 148 independent variables, 288 inertinite, 296 infrared absorption, 19 infrared active, 20 infrared detectors, 109 infrared inactive, 21 infrared light, 19 infrared linear dichroism, 203, 212 infrared microspectroscopy, 131 infrared spectra of a model compound for phospholipids, 260 infrared spectra of chlorophylls, 259 infrared spectra of intact bacterial cells, 263 infrared spectra of proteins, 255 infrared spectroscopy, 22

interferometer, 105 inversion centre, 47 irreducible representations, 71 isotope shift, 99 isotopic exchange, 202 Jacquinot advantage, 120 Johnson noise, 112 KBr method, 136 Krimm's rule, 202 Kubelka-Munk equation, 147, 150 Lambert-Beer's law, 131 Langmuir-Blodgett films, 146, 220, 236 LC/FT-IR, 172 light-emitting diodes, 216 linear dichroism, 203 linear momentum, 7 liquid crystal polymers, 211 liquid crystals, 206 liquid samples, 134 lithium cells, 221 loading plots, 294 low-temperature infrared spectroscopy, 255 MCT detector, 114, 188, 231 Mertz algorithm, 123 methacrylic acid, 218 methacryloxypropyltrimethoxysilane, 218 Michelson interferometer, 118 micro-crystal domains, 244 mirror misalignment, 124 mirrors, 107, 116 molar absorption coefficient, 133 molar absorptivity, 286 mole, 12 molecular electronics, 236 molecular orientation, 245 molecular vibration, 19, 23 353

monochromatic radiation, 12 Morse's function, 34 Mulliken symbols, 71 multiple linear regression, 290 multiplex advantage, 120 mutual exclusion rule, 30 Nernst glower, 107 noise equivalent power, 113 non-aqueous solutions, 133 normal coordinate, 20 normal frequency, 23 normal modes, 23 normal vibrations, 17, 23, 80, 97 normalized detectivity, 113 object space; 291 order, 43 order-disorder transition, 244 organic light emitting diodes, 214 organic thin films, 195 orientation measurements, 202 overfitting, 293 overtones, 21, 33, 37 parallel-polarized light, 145 partial least squares, 290, 294 particle theory, 1 penetration depth, 137 pentafluorophenylacrylate, 197 pentafluorophenylmethacrylate, 197 period of the motion, 3 perpendicular-polarized light, 144 phase transitions, 260 phospholipids, infrared spectra of a model compound for, 260 phosphonic acids, 166 photoacoustic signal, 155 photoacoustic spectroscopy (PAS), 131 photoacoustic techniques, 196 photoconductive effect, 111 photoelectromagnetic effect, 111 354

photoemissive effect, 111 photoinitiator, 218 photoisomerization, 220 photon, 6 photon detectors, 109,111 photosynthesis, 255 photovoltaic cells, 236 photovoltaic effect, 111 Planck's constant, 6 point groups, 53 polarizability, 79 polarized light, 144 poly(acrylic acid), 166 poly(ethylene glycol), 230 poly(ethylene oxide), 201 poly(methacrylic acid), 166 poly(methylmethacrylate), 165, 201 poly(propylene oxide), 197 poly(vinylphenol), 201 polyacrylonitrile, 166 polyatomic molecules, 29, 97 polyethylene, 206 polymer blends, 225 polymer characterization, 195 polymeric sulphonic acids, 166 polystyrene microspheres, 223 polyurethanes, 168 predictor variables, 288 principal axis, 43 principal component analysis, 290, 292 prisms and accessories for ATR spectroscopy, 139 product operation, 57 proper rotation axis, 43 proportionality constant, 286 propyl ester phosphazene, 166 protein dynamics, 255 proteins, infrared spectra of, 255 Pseudomonaschlororaphis,263 pyroelectric devices, 236 pyroelectric effect, 110 pyrolysis products, 182

quantitative analysis, 285 quantum, 6 quantum mechanics, 19 random coil structure, 256 Rayleigh radiation, 79 reducible representations, 70 redundant aperturing, 189 reflection absorption, 131, 143, 238, 241 refractive index, 2, 133 registration advantage, 120 response variables, 288 retardation, 119 retinals, 255 rocking vibrations, 30, 239 rotation reflection axis, 47 Schottky noise, 112 Schrodinger's equation, 27 scissoring, 30,239 score plots, 293 secondary structure, 142 selection rule, 19 SFC/FTIR, 182 silicone, 159 similarity transform, 65 simple harmonic, 2 size distribution, 151 spectrometer, 12 spectroscopic imaging instrument, 222 spectrum, 1 specular reflectance, 285 Staphylococcus aureus, 263 step-scan dynamic FT-IR, 209 step-Scan FT-IR, 125 step-scan impulse-response experiments, 210 Stokes lines, 79 stretching vibration, 23, 138 structural absorbance, 204 styrene-butadiene block copolymer blends, 165

subcell packing, 241 supercritical fluid chromatography, 182 surfactant, 184, 197 symmetric stretching, 239 symmetric stretching vibrations, 29 symmetry elements, 42 symmetry of a molecule, 20 symmetry operations, 42 symmetry plane, 47 synchronous correlation spectra, 321, 324 synchronous modulation, 125, 207

target projection plots, 306 target projections, 300 tetrachloroethylacrylate, 197 TGA/FT-IR, 158 thermal conductivity, 156 thermal detectors, 109, 110 thermogravimetric analysis, 302 thermoplastics, 163 thermopneumatic effect, 110 thermovoltaic effect, 110 thin films, 236 time-domain experiments, 125 time-resolved experiments, 125 time-resolved infrared spectroscopy, 255 total internal reflectance, 285 trans conformation, 248 transferrin receptor, 258 transition moment, 82,204 translation, 15 transmittance, 21,131 transmitted radiation, 12 trans-zigzagstructure, 240 triaminotriazine derivatives, 250 trisacrylates, 215 twisting vibrations, 30 two-dimensional correlation spectroscopy, 256, 318 355

univariate technique, 285 variable space, 291 vinylidene fluoride copolymers, 196 vinylidene fluoride-trifluorethylene, 201 vitrinite, 296 wagging vibrations, 30, 241 water molecules, 255

356

wave theory, 1,6 wavelength, 5 wavenumber, 5, 286 weathered sealants, 159 window materials, 133 x-ray diffraction, 319 zirconium oxide, 218