Spectroscopy: Atomic and Molecular 9789350434116, 9788183188098

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SPECTROSCOPY (ATOMIC AND MOLECUlAR) {For B.Sc. (Hons.) and M.Sc. Students of Indian Universities}

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SPECTROSCOPY (ATOMIC AND MOLECUlAR)

GORDEEP R. CHBnMAL

SHAM K. AnAnD

M.SC., Ph.D.

M. Sc. , Ph.D.

Reader in Chemistry

Reader in Chemistry

Dyal Singh College University of Delhi, Lodhi Road, . New Delhi-110003.

Dyal Singh College University of Delhi, Lodhi Road, New Delhi-110003.

Edited by:

M. ARORA ASEEM AnAnD

K4)]I GfIimaJaya GAlblishing'House MUMBAI Cl DELHI Cl BANGALORE Cl NAG PUR Cl HYDERABAD

© Author No part of this book shall be reproduced, reprinted or translated for any purpose whatsoever without prior permission of the Publisher in writing.

ISBN Revised Edition

Published by

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Contents o.

Concepts in Spectroscopy ......................................................•..................•...............•........... 1.1-1.56 0.1 Introduction to Spectroscopy ............................................................................................. 1.1 0.2 Properties of Electromegnetic Radiation .......................................................................... 1.2 0.3 Electromagnetic Spectrum ................................................................................................. 1.5 0.4 Different Types of Molecular Energies ............................................................................ 1.6 0.5 Interaction of Electromagnetic Radiation with Matter .................................................... 1.9 0.6 Molecular Absorption of Electromagnetic Radiation ..................................................... 1.10 0.7 Types of Molecular Spectra ............................................................................................ 1. 13 0.8 Emission of Radiant Energy by Atoms and Molecules ................................................ 1.14 0.9 Characteristics of Spectral Lines ..................................................................................... 1.16 0.10 Theoretical Principles of Atomic Spectroscopy ............................................................. 1.19 0.11 Theoretical Principles of Molecular Spectroscopy ......................................................... 1.50 0.12 Importance of Spectroscopy ............................................................................................ 1.50 0.13 Solved Examples .............................................................................................................. 1.51

1. Errors, 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8

Precision and Accuracy ............................................................................................ 2.1-2.6 Introduction ......................................................................................................................... 2.1 Type of Errors ................................................................................................................... 2.1 Significant Figures ............................................................................................................. 2.2 Precision and Accuracy ..................................................................................................... 2.2 Methods of Expressing Accuracy ..................................................................................... 2.2 Methods of Expressing Precision ...................................................................................... 2.3 Confidence Limits .............................................................................................................. 2.4 Photometric Errors ............................................................................................................. 2.5

2. Microwave Spectroscopy ...................................................................................................... 2.7-2.28 2.1 Introduction to Microwave Spectroscopy ......................................................................... 2.7 2.2 What is Microwave Spectroscopy ..................................................................................... 2.7 2.3 Differences between Infrared and Microwave Spectroscopy .......................................... 2.7 2.4 Theory of Microwave Spectroscopy ................................................................................. 2.8 2.5 Linear Molecules .............................................................................................................. 2.18 2.6 Spherical Top Molecules ................................................................................................. 2.18 2.7 Symmetric Top Molecules ............................................................................................... 2.18 2.8 Asymmetric Top Molecules ............................................................................................ 2.20 2.9 The Stark Effect ............................................................................................................... 2.20 2.10 Instrumentation for Microwave Spectroscopy ................................................................ 2.21 2.11 Applications of Microwave Spectroscopy ...................................................................... 2.23

(vi)

3. Infrared Absorption Spectroscopy ••••.•.•.•.••••••••.••••••••••.•••••••....•••••.••...••••••••..•..•••••••.•....•••.• 2.29-2.82 3.1 Introduction ....................................................................................................................... 2.29 3.2 The Range of Infrared Radiation .................................................................................... 2.30 3.3 Nomenculature of Infrared Spectra ................................................................................. 2.30 3.4 Theory of Infrared Absorption Spectroscopy or Requirements for Infrared Radiation Absorption ......................................................................................... 2.31 3.5 Mathematical Theory of IR Absorption Spectroscopy ................................................. 2.32 3.6 Linear Molecules .............................................................................................................. 2.38 3.7 Symmetric Top Molecules ............................................................................................... 2.39 3.8 Asymmetric Molecules .................................................................................................... 2.40 3.9 Instrumentation ................................................................................................................. 2.40 3.10 Single Beam and Double Beam Spectrophotometers .................................................... 2.51 3.11 Modes of Vibrations of Atoms in Polyatomic Molecules ............................................ 2.52 3.12 Factors Which Influence Vibrational Frequencies ......................................................... 2.55 3.13 Selection Rules ................................................................................................................. 2.58 3.14 Position and Intensity of Bands ...................................................................................... 2.59 3.15 Intensity of Absorption Bands ........................................................................................ 2.60 3.16 Units of Measurement ..................................................................................................... 2.61 3.17 Application of Infrared Spectroscopy to Organic Compounds ..................................... 2.62 3.18 Applications of Infrared Spectroscopy to Inorganic Complexes .................................. 2.73 3.19 Miscellaneous Examples .................................................................................................. 2.75 3.20 Attenuated Total Reflectance .......................................................................................... 2.76 3.21 Nondispersive Infrared ..................................................................................................... 2.77 3.22 Photothermal Beam Deflection Spectroscopy (PBDS) .................................................. 2.78 3.23 Application of Infrared Spectroscopy to Quantitativ~ Analysis ................................... 2.78 3.24 Limitation of Infrared Spectroscopy ............................................................................... 2.80 4. Raman 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9

Spectroscopy ......................................................................................................... 2.83-2.106 Introduction ....................................................................................................................... 2.83 Principle ............................................................................................................................ 2.83 Characteristic Properties of Raman Lines ...................................................................... 2.85 Differences between Raman Spectra and Infrared Spectra ........................................... 2.85 Mechanism of Raman Effect .......................................................................................... 2.86 Instrumentation ................................................................................................................. 2.92 Intensity of Raman Peaks ................................................................................................ 2.98 Applications of Raman Spectroscopy ............................................................................. 2.99 Short Type Questions .................................................................................................... 2.105

5. Visible 5.1 5.2 5.3 5.4· 5.5 5.6 5.7 5.8

Spectrophotometry and Colorimetry ............................................................... 2.107-2.148 Introduction ..................................................................................................................... 2.107 Theory of Spectrophotometry and Colorimetry ........................................................... 2.108 Deviations from Beer's Law ......................................................................................... 2.112 Instrumentation ............................................................................................................... 2.116 Obtaining and Interpreting Data ................................................................................... 2.131 Applications of Colorimetry and Spectrophotometry .................................................. 2.135 Molar Composition of Complexes ................................................................................ 2.135 Spectrophotometric Titrations ........................................................................................ 2.142

(vii)

6. Ultraviolet Spectroscopy ................................................................................................. 2.149-2.184 6.1 Introduction ..................................................................................................................... 2.149 6.2 Origin and Theory of Ultraviolet Spectra .................................................................... 2.149 6.3 Types of Transitions of Inorganic Molecules .............................................................. 2.l50 6.4 Types of Transitions in Organic Molecules ................................................................. 2.l51 6.5 The Shape of UV Absorption Curves .......................................................................... 2.155 6.6 Transition Probability ..................................................................................................... 2.156 6.7 Chromophore and Related Terms ................................................................................. 2.l56 6.8 Effect of Conjugation .................................................................................................... 2.160 6.9 Solvent Effects ............................................................................................................... 2.160 6.10 Choice of Solvent .......................................................................................................... 2.161 6.11 Woodward-Feiser Rules for Calculating Absorption Maxima ................................... 2.l62 6.12 Instrumentation ............................................................................................................... 2.167 6.13 Applications of Spectroscopy to Organic Compounds ................................................ 2.172 6.14 General Applications of OV Absorption Spectroscopy ............................................... 2.177 7. Nuclear Magnetic Resonance ......................................................................................... 2.185-2.234 7.1 Introduction to Nuclear Magnetic Resonance .............................................................. 2.185 7.2 Quantum Description of Nuclear Magnetic Resonance .............................................. 2.186 7.3 Rules Predicting spin Numbers of Nuclei and Calculation of spin Numbers of Elements Responding to NMR ................................................................................ 2.187 7.4 Width of Absorption Lines in NMR ............................................................................ 2.l88 7.5 Number of Signals: Equivalent and Non-Equivalent Protons .................................. 2.l90 7.6 Chemical Shift ................................................................................................................ 2.192 7.7 Chemical Shifts of Different Types of Protons and Positions of PMR Signals ...... 2.l96 7.8 Spin-Spin Coupling: Splitting of Signals ................................................................... 2.l98 7.9 Coupling Constants ........................................................................................................ 2.202 7.10 Instrumentation ............................................................................................................... 2.203 7.11 Relationship Between the Area of the Peaks and Molecular Formula ...................... 2.210 7.12 Solvents Used in NMR ................................................................................................. 2.212 7.13 Interpretation of NMR Spectra ..................................................................................... 2.212 7.14 Applications of NMR Spectroscopy ............................................................................. 2.220 7.15 Limitations of NMR Spectroscopy ............................................................................... 2.231 7.16 Fluorine-9 NMR ............................................................................................................ 2.231 7.17 Phosphorus-31 NMR ..................................................................................................... 2.231 7.18 Carbon-13 NMR ............................................................................................................ 2.231 8. Nuclear Magnetic Double Resonance (NDMR) and INDOR Spectroscopy .•.•.•..... 2.235-2.237 8.1 Introduction ..................................................................................................................... 2.235 8.2 Spin-Spin Decoupling by NDMR ................................................................................. 2.235 8.3 INDOR Spectroscopy ..................................................................................................... 2.237 9. NQR Spectroscopy ........................................................................................................... 2.238-2.244 9.1 Introduction ..................................................................................................................... 2.238 9.2 Theory of NQR .............................................................................................................. 2.238 9.3 Instrumentation for NQR Spectroscopy ........................................................................ 2.241 9.4 Sample Requirements ..................................................................................................... 2.242 9.5 Applications of NQR ..................................................................................................... 2.242

(viii)

10. Electron Spin Resonance Spectroscopy ........................................................................ 2.245-2.271 10.1 Introduction ..................................................................................................................... 2.245 10.2 Comparison Between NMR and ESR .......................................................................... 2.245 10.3 Historical Facts ............................................................................................................... 2.245 10.4 Types of Substances with Unpaired Electrons ............................................................ 2.246 10.5 Theory of ESR ............................................................................................................... 2.246 10.6 Instrumentation ............................................................................................................... 2.247 10.7 Presentation of the ESR Spectrum ............................................................................... 2.251 10.8 Hyperfine Splitting ......................................................................................................... 2.252 10.9 Determination of g-Value ............................................................................................. 2.255 10.10 Deviation of the Value of g ......................................................................................... 2.255 10.11 Line Width ..................................................................................................................... 2.256 10.12 Applications of ESR Spectroscopy ............................................................................... 2.259 10.13 ENDOR ........................................................................................................................... 2.267 10.14 ELDOR ........................................................................................................................... 2.268 11. Mass Spectrometry .......................................................................................................... 2.272-2.302 11.1 Introduction ..................................................................................................................... 2.272 11.2 Theory ............................................................................................................................. 2.272 11.3 Components of Mass Spectrometer .............................................................................. 2.274 11.4 Recording of a Mass Spectrogram ............................................................................... 2.284 11.5 Resolution of Mass Spectrometer ................................................................................. 2.284 11.6 Types of Ions Produced in a Mass Spectrometer ....................................................... 2.285 11. 7 General Rules for Interpretation of Mass Spectra ....................................................... 2.287 11.8 A Typical Example of Interpretation of Molecular Mass Spectra ............................. 2.288 11.9 Some Example of Mass Spectra ................................................................................... 2.293 11.10 Quantitative Analysis ..................................................................................................... 2.295 11.11 Applications of Mass Spectrometry .............................................................................. 2.296 12. X-Ray Absorption, Diffraction, and Fluorescence Spectroscopy (Crystal Tonography) ...................................................................................................... 2.303-2.339 12.1 Introduction ..................................................................................................................... 2.303 12.2 General Theory ............................................................................................................... 2.303 12.3 Instrumentation ............................................................................................................... 2.309 12.4 X-ray Absorption Apparatus ......................................................................................... 2.314 12.5 Non-Dispersive X-ray Absorption Method ................................................................. 2.314 12.6 Applications of X-ray Absorption Methods ................................................................ 2.315 12.7 X-ray Diffraction Methods ........................................................................................... 2.318 12.8 Applications of X-Ray Diffraction ............................................................................... 2.326 12.9 Fluorescence Methods .................................................................................................... 2.332 12.10 Crystal Tonograpy .......................................................................................................... 2.338 13. Atomic 13.1 13.2 13.3 13.4 13.5

Absorption Spectroscopy .................................................................................. 2.340-2.366 Introduction ..................................................................................................................... 2.340 Principle .......................................................................................................................... 2.340 Grotrian Diagrams .......................................................................................................... 2.341 Detection of Non-metals by Atomic Absorption Spectroscopy .................................. 2.342 Differences Between Atomic Absorption Spectroscopy and Flame Emission Spectroscopy ....................................................................................... 2.342

13.6 Advantages of Atomic Absorption Spectroscopy Over Emission Flame Spectroscopy ....................................................................................... 2.343 13.7 Disadvantages of Atomic Absorption Spectroscopy .................................................... 2.343 13.8 Instrumentation ............................................................................................................... 2.344 13.9 Operation of Atomic Absorption Spectrometer ........................................................... 2.354 13.10 Single and Double Beam Atomic Absorption Spectrometer ...................................... 2.354 13.l1 Detection Limit and Sensitivity .................................................................................... 2.355 13.12 Interference ..................................................................................................................... 2.355 13.13 Absorption and Emission Line Profile ......................................................................... 2.359 13.14 Typical Analyses ............................................................................................................ 2.360 13.l5 Applications of Atomic Absorption Spectroscopy ....................................................... 2.360

14. Flame 14.1 14.2 14.3 14.4 14.5 14.6 14.7 14.8 14.9 14.10 14.11 14.12 14.13

Photometry and Flame Infrared Emission (FIRE) ........................................ 2.367-2.388 Introduction ..................................................................................................................... 2.367 Limitations of Flame Photometry ................................................................................. 2.367 General Principles of Flame Photometry ..................................................................... 2.367 Instrumentation ............................................................................................................... 2.370 Effect of Solvent in Flame Photometry ....................................................................... 2.376 Instruments ...................................................................................................................... 2.376 Application of Flame Photometry ................................................................................. 2.378 Interferences in Flame Photometry ............................................................................... 2.380 Factors that Influence the Intensity of Emitted Radiation in a Flame Photometer .. 2.382 Limitations of Flame Photometry ................................................................................. 2.382 Determination of Non-metals ....................................................................................... 2.382 Flame Infrared Emission (FIRE) ................................................................................... 2.386 Conclusions ..................................................................................................................... 2.386

15. Nephelometry and Turbidimetry ••..•.•.•.•.•...........•...•.•.•.......•.•..•.•..•.••••.•..•.•.•..••.•.•..•.••..•• 2.389-2.398 15.1 Introductory ................................................. :.................................................................. 2.389 15.2 Turbidimetry and Colorimetry ....................................................................................... 2.389 15.3 Nephelometry and Fluorimetry ..................................................................................... 2.390 15.4 Choice Between Nephelometry and Turbidimetry ....................................................... ~.390 15.5 Theory ............................................................................................................................. 2.390 15.6 Comparison of Spectrophotometry, Turbidimetry and Nephelometry ........................ 2.392 15.7 Instrumentation ............................................................................................................... 2.392 15.8 Applications of Turbidimetry and Nephelometry ........................................................ 2.395 16. Fluorimetry and Phosphorimetry .................................................................................. 2.399-2.416 16.1 Introduction ..................................................................................................................... 2.399 16.2 Comparison of Absorption and Fluorescence Methods ............................................... 2.399 16.3 Theory ............................................................................................................................. 2.400 16.4 Instrumentation ............................................................................................................... 2.405 16.5 Applications of Fluorimetry .......................................................................................... 2.411 16.6 Applications of Phosphorimetry ........•........................................................................... 2.414 16.7 Comparison of Fluorimetry and Phosphorimetry ......................................................... 2.415 16.8 Comparison of Fluorimetry and· Phosphorimetry with Absorption Methods ............. 2.415 17. Mossbauer Spectroscopy ............................•.•....•.•.•..•......•...•....•.•.••.•...................••.•....•.•• 2.417-2.427 17.1 Principle .......................................................................................................................... 2.417 17.2 Instrumentation and Mossbauer Spectra ....................................................................... 2.418 17.3 Applications .................................................................................................................... 2.422

(x)

18. Emission Spectroscopy .................................................................................................... 2.428-2.448 18.1 Introductory .................................................................................................................... 2.428 18.2 Theory ............................................................................................................................. 2.428 18.3 Instrumentation ............................................................................................................... 2.430 18.4 Spectrographs .................................................................................................................. 2.436 18.5 Application of Emission Spectroscopy ......................................................................... 2.439 18.6 Advantages and Disadvantages of Emission Spectroscopy ......................................... 2.446 19. Refractometry ................................................................................................................... 2.449-2.457 19.1 Introduction ..................................................................................................................... 2.449 19.2 Abbe Refractometer ....................................................................................................... 2.452 19.3 Applications of Refractometry .................................... :................................................. 2.455 19.4 Optical Exaltation ........................................................................................................... 2.457 20. Polarimetry ....................................................................................................................... 2.458-2.467 20.1 Introduction ..................................................................................................................... 2.458 20.2 Plane Polarised Light ..................................................................................................... 2.458 20.3 Optical Activity .............................................................................................................. 2.460 20.4 Types of Molecules Analysed by Polarimetry ............................................................. 2.461 20.5 Theory of Optical Activity ............................................................................................ 2.461 20.6 Polarimeter ...................................................................................................................... 2.462 20.7 Applications of Optical Activity ................................................................................... 2.464 21. Optical 21.1 21.2 21.3 21.4 21.5 21.6 21.7 21.8 21.9 21.10 21.11

Rotatory Dispersion and Circular Dichroism Spectroscopy ....................... 2.468-2.481 Theory: Polarised Light ............................................................................................... 2.468 Optically Active Molecules ........................................................................................... 2.470 Optical Rotatory Dispersion .......................................................................................... 2.471 Circular Dichroism ......................................................................................................... 2.472 Cotton Effect .................................................................................................................. 2.473 Octant Rule ..................................................................................................................... 2.473 Faraday and Kerr Effects .............................................................................................. 2.475 Instrumentation ..................................................................... ~......................................... 2.475 Automatic Recording Spectropolarirneters .................................................................... 2.476 Instruments for Circular Dichroism Measurement.. ..................................................... 2.478 Applications of Optical Rotatory Dispersion and Circular Dichroism ....................... 2.478

22. Atomic 22.1 22.2 22.3 22.4 22.5

Fluorescence Spectroscopy (AFS) ..••.•.......•••...•....•••••••.....•.•••••.••.•......•.•....•.••...••••.. 3.1-3.17 Introduction ................................... :.................................................................................... .3.1 Mathematical Relationships ............................................................................................... 3.3 Advantages of Atomic Fluorescence ................................................................................ 3.5 Limitations of Atomic Fluorescence Spectroscopy .......................................................... 3.5 Atomic Fluorescence Spectroscopy as an Analytical Tool ............................................. 3.6

23. Photoelectron Spectroscopy ................................................................................................ 3.18-3.26 23.1 Introduction .............................................................................................................. ,........ 3.18 23.2 Principle of Photoelectron Spectroscopy ........................................................................ 3.18 23.3 Instrumentation ................................................................................................................. 3.19 23.4 Theory of Photoelectron Spectroscopy ........................................................................... 3.21 23.5 Applications of Photoelectron Spectroscopy .................................................................. 3.24 23.6 Comparison with Other Methods .................................................................................... 3.25

(xi) 24. Electronic Spectroscopy .......................... 3.27-3.43 24.1 Introduction .. :.................................................................................................................... 3.27 24.2 Frank Condon Principle ................................................................................................... 3.28 24.3 Vibrational Coarse Structure of Electronic Spectra ....................................................... 3.31 24.4 Fortrat Diagram ................................................................................................................ 3.34 24.5 Electronic Spectra of Transition Metal Ions .................................................................. 3.35 24.6 Electonic Spectra of Organic Molecules ........................................................................ 3.36 24.7 Charge-Transfer (CT) Spectra ........................................................................................ 3.38 24.8 Electronic Spectra of Conjugated Molecules ................................................................. 3.39 24.9 Application of Electronic Spectra to Transition Metal Complexes .............................. 3.42 m

.........................................................................

25. Reflectance Measurements or Reflectance Spectroscopy ............................................... 3.44-3.45 25.1 Introduction ....................................................................................................................... 3.44 25.2 Instruments, Techniques and Applications ..................................................................... 3.44 25.3 Applications of Reflectance Measurements .................................................................... 3.44

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o Concepts in Spectroscopy 0.1 Introduction to Spectroscopy The spectroscopic techniques form the largest and most important single group of techniques used in chemistry. These techniques provide a wide range of qualitative and quantitative information. All spectroscopic techniques are more or less dependent on the emission or absorption of electromagnetic radiation characteristic of certain energy changes within an atomic or molecular system. The energy changes are associated with a complex series of discrete or quantised energy levels in which atoms and molecules are assumed to exist. The most important use of spectroscopic techniques in analytical chemistry is in the determination of atomic and molecular structure, including functional groups and stereochemical arrangements. Different types of spectroscopic techniques are currently In wide use because they require a very small amount of material and time in analysis. Spectroscopy and its applications form a significant part of modem chennstry and physics. From its derivation the word spectroscopy appears to mean the watching of images, but the modem subject covers the interaction of electromagnetIc radiations with matter. The most important consequence of such interaction is that energy is absorbed or emitted by the matter in discrete amounts called quanta. The absorption or emission processes are known throughout the electromagnetic spectrum ranging from the gamma region (nuclear resonance absorption or the Mossbauer effect) to the radio region (nuclear magnetic resonance) When the measurement of radiation frequency is done experimentally, it gives a value for the change of energy involved and from this one may draw the conclusion about the set of possible discrete energy levels of the matter. The ways in which the measurements of radiation ji-equency (emitted or absorbed) are made experimentally and the energy levels deduced from these comprise the practice of spec~roscopy.

The various branches of spectroscopy generally involve measurements of two important experimental parameters which are the energy of the radiation absorbed or emitted by the system and the intensity of the spectral lines. In most branches of spectroscopy the system interacts with the electric fif'ld. However, in case of magnetic resonance spectroscopy it interacts with the magnetic field. Spectroscopy is one of the most powerful tools available for the study of atonnc and molecular structure and is used in the analysis of a wide range of samples. The study of spectroscopy can be carried out under the following heads : (aJ Atomic Spectroscopy : It deals with the interaction of electromagnetic radiation with atoms which are most commonly in their lowest energy state called the ground state. Monoatomic substanc.!s normally exist in the gaseous state and are able to absorb electromagnetic radiation, resulting in transitions of electrons from one electronic energy level to another. The electronic absorption of electromagnetic radiation can occur only if the photon has an energy WhICh is equal to the energy difference between two quantized energy levels, i.e., , , L1E = hv

1.2

Concepts in Spectroscopy

where 6.E is the energy difference between two quantum levels and v is the frequency of photon which can result in the electronic excitation. Applications of atomic spectroscopy in the field of chemistry are few. However, its importance has increased by the development of lasers. (b) Molecular Spectroscopy : It deals with the interaction of electromagnetic radiation with molecules. This results in transitions between rotational and vibrational energy levels in addition to electronic transitions. As a result, the spectra of molecules are much more complicated than those of atoms.

In contrast to atomic spectra which arise from the transitions of an electron between the atomic energy levels, the molecular spectra arise from three types of transitions, viz., rotational, vibrational and electronic transitions. Molecular spectra extend from the visible through infrared into the microwave region. Current interest in molecular spectroscopy is very great because the number of known molecules are extremely large as compared with free atoms. From molecular spectra, information may be obtained about molecular vibrations and rotations that reveal a great deal about molecular structure. The detailed information regarding molecular structure (molecular symmetry, bond distances and bond angles) and chemicals properties (electronic distribution, bond strength and intra-and inter-molecular processes) c.an be obtained from the atomic and molecular spectra. The spectroscopic investigations were first carried out by Newton in the seventeenth century, the quantitative treatment of the subject was possible only after the introduction of quantum mechanics. Initially, it was atomic spectra that provided the first direct experimental evidence for the various quantum mechanical postulates. 0.2 Properties of Electromagnetic Radiation Introduction : Electromagnetic radiation is a form of energy that is transmitted through space at an enormous velocity. Electromagnetic radiation requires no supporting media; in fact it more readily passes in a vacuum than in a supporting medium such as air. In contrast to ·this, other waves, such as sound, do require supporting medium. The most obvious example of electromagnetic radiation is light which occupies only a small region in a spectrum of electromagnetic radiation. An electromagnetic radiation is said to have a dual nature, exhibiting both wave and particle characteristics. This duality is not confined to the visible portion of the electromagnetic spectrum but can be demonstrated for the whole region of electromagnetic radiation. Indeed this duality is useful for the quantitative description of many phenomena. Wave Properties of Electromagnetic Radiation: As the name implies, an electromagnetic radiation is an alternating electrical and associated magnetic force in space. Thus, an electromagnetic wave has an electric component and magnetic component. The two components oscillate in planes perpendicular to each other and perpendicular to the direction of propagation of the radiation. Fig. 0.1 is a vector representation of electromagnetic radiation moving along the x-axis. The electric field E varies in the direction of y axis, and the corresponding magnetic field H varies in the direction of z-axis. For radiation travelling in x direction in free space these vary with time and distance as follows: Ey cos 21t (vt-x/"}.,) and Hz cos 21t (vt-x/"}.,) The subscripts y and z indicate that the electric and magnetic fields are mutually perpendicular to each other and to the direction of propagation, x. If the whole beam of radiation is of the above form

Concepts in Spectroscopy

1.3 y

Electric Vector

--:of-----:+----1:---_~,.:;x.. Direction of Propagation

z

Magnetic Component

Fig. 0.1 : Representation of an Electromagnetic Wave

it is said to be coherent and plane polarized. Two beams or two parts of the same beam, are said to be incoherent if the phase of one is unrelated to that of the other. In most of the cases, the various properties of radiation can be described by changing only the electric vector. But in a few cases, notably in the discussion of magnetic resonance, it is more convenient to deal with the magnetic vector. The velocity of electromagnetic radiation in vacuum is independent of frequency and has the value of 3xl08m per second. In contrast to other wave phenomena such as sound, however, electromagnetic radiation does not require any supporting medium for its transmission, and it readily passes through vacuum. An electromagnetic wave, such as that of Fig. 0.1, is characterised by the following parameters : (a) Wavelength, A : It is the distance between two successive maxima on an electromagnetic wave. It is denoted by the Greek letter Lambda A. The usual units of wavelength vary with the region of electromagnetic spectrum. Some of more commonly employed units are meters (m), centimeters (cm). millimeters (mm) , micrometers ()1m), nanometers (nm) and angstrom A. These units are related by the following relations : 1 11m = lO-4cm = 1O-3 nm = 1O-6m 1 nm = 1O-6)1m = 1O-7cm = 1O-9m = lOA 1 A = 1O-8cm = O.lnm = 1O-lcm The nanometer is also designated as millimicron (mil). The term nanometer follows the recommendations of the National Bureau of Standards in 1963. A beam carryirIg radiation of only one discrete wavelength is said to be monochromatic, and a beam having radiation of several wavelengths is said to be polychromatic or heterochromatic. (b) Frequency, v : The number of complete wavelength units passing through a given point in unit time is called the frequency of radiation. It is denoted by the Greek letter, v. Frequency has. unit. of reciprocal time. It is generally expressed in cycles per seconds or in Hertz (Hz) or in Fresnel. 1 Hz = 1 cycle s-I 1 Fresnel = 10 12 Hz The other units of frequency are kilocycles per second (k cps or k Hz) and megacycles per second (M cps or MHz). IMHz = 103 kHz = 106Hz

v:

(c) Wave Number, Though the frequency is more fundamental than the wavelength, it is cumbersome to use in practice, because it is usually a very large number. It is a common practice, therefore, to express the frequency as the wave number which is defIned as the number of waves per centimeter

in vacuum. This quantity is generally denoted by

v.

1.4

Concepts in Spectroscopy 1

v=-

(0.1)

A,

Wave number has units of reciprocal distance (cm- I ) for which the name Kaiser has been abbreviated (K). Sometimes kilokayser is also used (kK). 1 kK = 1000K = 1000 cm- I The wave number notation is generally used in infrared spectroscopy. (d) Velocity, v : In constant to frequency, the velocity is dependent upon the medium through which radiation is passing. It has the unit of cm s-I or ms-I. The product of wavelength and frequency is equal to the velocity of the wave in the medium, i.e., wavelength x frequency = velocity Ax v = v or As the density of the medium through which electromagnetic radiation is passing decreases, the wavelength and velocity of radiation increase, reaching constant maximum values in vacuum. At this point the velocity is 3x10 l ocm s-I. Since the frequency must remain constant, the relationship between the wavelength, frequency, and velocity of light in a vaccum is AV = c (e) Relation between Frequency, Velocity and Wavenumber: Recalling the definitions offrequency, wavelength and wave number, one may write alternative formulation of Eq. (0.1). 1

Thus,

v=-

But

v = ~ or A,

A,

A,

=::... v

v c

v=-

or

v

(0.2)

=cv

(0.3) (3 x 10 8m s-I).

where c is the velocity of light in a vacuum Particle Properties of Electromagnetic Radiation : The phenomena of refraction, reflection, reinforcement and destructive interference are examples of wave properties. But the wave nature of electromagnetic radiation fails to explain many phenomena like photoelectric effect, etc. In order to explain such phenomena, it is assumed that the electromagnetic radiation consists of a stream of discrete packets (particles) of pure energy, called photons or quanta. These have definite energy and travel in the direction of propagation of the radiation beam with the velocity equal to that of the light. The energy of the photon is proportional to the frequency of radiation and is given by the relationship E = hv

(0.4)

where E is the energy of photon in ergs, v the frequency of the electromagnetic radiation in cycles per second and h is called Planck's constant and has a value of 6.624x 10-27 erg-second or 6.624x 10-34 Joule-second in SI units. A photon of low frequency (longer wavelength) has a lower energy content than one of higher frequency (shorter wavelength). The intensity of a beam of radiation is proportional to the number of photons per second that are propagated in the beam. But the intensity is independent of the energy of each photon. Relation Between Wavelength and Particle Properties of Electromagnetic Radiation : The connection between the wave and particle characteristics of radiation was suggested by Planck. According to him, if a transition occurs between the energy states of a system such that a photon of energy E is emitted or absorbed, the frequency, v, of the emitted or absorbed radiation is given by

Concepts in Spectroscopy

1.5

hv (0.4) Recalling the definitions of frequency, wavelength and wavenumber from the proceeding section, one may write alternative formulation of Eq. 0.4 that relates the photon energy to the wavelength or wave number of radiation : E

=

- he E =hv=hc V=(0.5) A With the help of this equation, a particular wavelength or wave number of frequency can be converted into desired units (see solved examples). It is customary to assume at the present time, the validity of both the theories, i.e., the wave theory and the particle theory and interpret results making use of either or both the theories.

0.3 Electromagnetic Spectrum The entire range over which electromagnetic radiation exists is known as electromagnetic spectrum. The electromagnetic spectrum covers an immense range of wavelength. Fig. 0.2 gives a diagrammatic representation of the electromagnetic spectrum. A logarithmic scale has been employed in this representation. The divisions between the different spectral regions are a matter of convenience and are made partly to indicate the origin of radiation and partly for experimental (instrumental) reasons. The limits indicated in Fig. 0.2 are arbitrary and diffuse-the regions overlap. 10'"

gamma

10.9

Violet, Indigo Blue Green Yellow

10.7

..,

10.5

E

3 S. 10. ~

c;, 10" c: >

~

Infrared Microwaves

Q)

CD

_--:-:,....,.....,..400

Orange Red

10' 10

800

3

10

5

Radio Waves

7

10 10

9

Fig. 0.2 : The electromagnetic spectrum. The visible portion is shown on an expanded scale to the right.

The major characteristics of various spectral regions are outlined as follows : (aJ rray Region: This lies between 0.2 to 1 A. The gamma rays are shortest waves emitted by atomic nuclei, involving energy changes of 109 to 1011 Joules/gram atom. (b) X-ray Region : This lies between 1 to lOA. X-rays emitted or absorbed by movement of electrons close of nuclei of relatively heavy atoms, involve energy changes of the order of ten thousand kilo Joules. (c) Visible and Ultraviolet Region : These are further made up of the following regions : Vacuum ultraviolet 1-180 nm

Concepts in Spectroscopy

1.6

180-400 run Ultraviolet 400-750 run Visible The distinction between "vacuum ultraviolet" and "ultraviolet" is made because air starts absorbing below 180 nanometers, and so does the quartz which is generally used to construct cells and windows. This fact makes complications in the task of observing and measuring radiations below 180 run in wavelength, and indeed this region of the spectrum may be regarded as the beginning for chemical analysis. In near future, this may find some valid applications in chemical analysis. Instruments are now available for use in the vacuum ultraviolet region. Radiation which we call light form the visible portion of the electromagnetic spectrum. This region is approximately 400-750 run in wavelength. Within the visible region of the spectrum, persons with normal colour vision are able to correlate the wavelength of light striking the eye with the subjective sensation of colour. The colours oflight correspond to different wavelengths (see Table 0.1). The ranges of wavelengths given in this table are approximate because the colour shades into another throughout the electromagnetic spectrum. In order to measure the wavelengths in ultraviolet and visible regions we require several different units of measurements. Table 0.1 : Colours of Visible Light Colour Violet Blue Green-Blue Blue-Green Green YeJlow-Green Yellow Orange Red

Wavelength, nm

400-435 435-480 480-490 490-500 500-560 560-580 580-595 595-610 610-750

(d) Infrared Region: This region has been further divided into the following sub-regions:

0.7-2.5 Jl Infrared (near) Infrared 2.5-15 Jl Far infrared 15-200 Jl All the three sub-regions of infrared part of the electromagnetic spectrum are associated with the changes in the vibration of molecules, and the distinction between them is a simple matter of instrumentation. The infrared region is one of the most valuable regions for the chemist. (e) Microwave Region : (0.1 mm to 1 cm wavelength) : This region corresponds to changes in the rotation of molecules. Separations between the rotational levels of molecules are of the order of hundreds of Joules per mole. (For details, see the chapter on "Microwave Spectroscopy.") (f) Radio Frequency Region: (10m--l cm wavelength) : The energy change invoived in this region arises due to the reversal of a spin of nucleus or electron. This is of the order 0.001-10 Joules/ mole. (For details see the chapter on "Nuclear Magnetic Resonance.")

0.4 Different Types of Molecular Energies In addition to the ordinary energy of translational motion which is not of concern in molecular srectroscopy, a molecule may possess internal energy which can be subdivided into three classes : rotational energy, vibrational energy and electronic energy. (0.6)

Concepts in Spectroscopy

1.7

It will be seen further that Eelec>EVlb>Eror The laws of quantum energy which are applicable to the atomic spectra are also applicable to molecular spectra. Thus, on the basis of quantum theory, the three components of the molecular energy can have only discrete and quantised values.

In order to understand the concepts of electronic energy, vibrational energy and rotational energy, it is desirable to study them along with translational energy. Let us discuss these one by one. (a) Translational Energy : This energy is associated with the uniform motion (velocity) of a molecule as a whole. This motion is generally described with respect to the centre of mass of the molecule (Fig. 0.3). The classical energy due to translational motion is given by Et =

Yzmv 2

(0.7)

where Et is the translational energy of molecular mass m moving with velocity v with respect to the centres of mass. Due to translational motion, the molecule is free to move in the three perpendicular directions x, y. z. It means that it has three degrees of freedom. But the translational energy of a molecule of gas by the classical theorem of the equipartition of energy along any axis is Y:zkT or YzRT per mole. Therefore, the translational energy of molecule of a gas along three mutually perpendicular axes will be 3 x Y:zkT or 312 kT or 312 RT per mole. In solids, the translational freedom is restricted. According to the quantum mechanical equations of translational motion of an isolated molecule of mass M in a rectangular box of dimension a xb xc, the value of translational energy is given by

(0.8) where the values of n are restricted to integer values. As the number of translational energy levels is exceedingly large and the difference between them is negligible, it means that continuous range of translational energies are available. The concept of translational energy is not of much use in molecular spectroscopy.

(b) Rotational Energy: This type of energy is associated with the overall rotation of the molecule with atoms considered as fixed point masses. According to the classical theory, the value of a rotational energy is given by (0.9) where I is the moment of inertia and w the angular velocity of rotating molecule. However, the quantum mechanical formula for the rotational energy of a simple linear molecule is given by

h2

E ret = J(J + 1)-287t I

(0.10)

where I is the moment of inertia of a simple linear molecule and J is zero or a positive integar called the rotational quantum number. Each rotational level has a (21+ 1) fold degeneracy. Equation (0.10) does not apply in the liquid phase where molecular collisions are frequent. It is important to remark here that only the polar molecules can rotate in an electromagnetic field.

A monoatomic molecule possesses only one rotational degree of freedom. But a non linear tri-orpoly-atomic molecule may rotate about the three perpendicular axes passing through the centre of gravity (Fig. 0.3). It is therefore said to have three rotational degrees of freedom. A linear molecule cannot rotate about the axis passing through the nuclei of the bonded atoms. It means that one of its rotational degrees of freedom is restricted. Hence a linear molecule possesses only two rotational degrees of freedom. It is to be noted that equation (0.10) does not apply in the liquid phase where molecular collisions are frequent. Instead the rotation is better treated in a classical manner analogous to translations. In most

1.8

Concepts in Spectroscopy

Vibrational

Fig. 0.3 : Three different ways in which molecules can store energy as a result of motion of atoms in a non-linear triatomic molecule.

of the crystalline solids free rotation is prevented and replaced by an oscillatory motion which can be treated as a vibration. (c) Vibrati9nal Energy : This type of energy is associated with the oscillation of atoms (of the molecule) which are considered as point masses about equilibrium positions. This type of energy can be treated on a quantum mechanical basis. The vibrational energy of a molecule may be given as

EV'b = hv (v + V,) (0.11) where v is the vibrational frequency and v is zero or a possive integer called the vibrational quantum number. It is to be noted that the form of equation (0.11) does not fulfil the convention that the energy shall be zero for the ground state. The ground state corresponds to v = 0 with an energy hv/2, this is generally known as zero point energy. The origin corresponds to a molecule with all its nuclei at their equilibrium positions and devoid of relative motion. At room temperature v = 1 and v = 2 states of low frequency vibration are appreciably populated. It should be kept in mind that for the higher frequency vibrational degrees of freedom for all molecules have v = 0 at equilibrium. The vibrational degrees of freedom for non-linear molecules such as H20, N02 and CH4 are 3n-6. But there are 3n-5 vibrational degrees of freedom for linear molecules, e.g., CO 2, C2H2 and 13-, When a molecule absorbs energy, the increase in vibration motion of the molecule is usually accompanied by increased rotation of the same molecule. This combination provides the basis for IR absorption spectroscopy. (d) Electronic Energy : This type of energy is associated with the 'motion' of electrons while considering the nuclei of atoms (of a molecule) as fixed points. The increase in the electronic energy of a molecule occurs due to an increase in the kinetic energy required to move the electron from the ground state to an excited state and varies with the type of molecular bond in which the electron exists. Further, the vibrational and rotational 'energies of the molecules are added to the electronic energies.

Concepts in

~pectroscopy

1.9

Different electronic states may differ appreciably in their associated momentum of inertia and vibrational frequencies so that the permitted values of Era! and EVlb are considerably altered by electronic excitation. There is no general formula for electronic energies in terms of simple quantum numbers, except for hydrogen like atom.

0.5 Interaction of Electromagnetic Radiation with Matter Absorption of Radiation : When electromagnetic radiation passes through matter, a variety of phe~omena riiay occur. Some of these are as follows : (1) If the photons of radiation possess the appropriate energies, they may be absorbed by the matter and result in electronic transitions, vibrational changes or rotational changes or combination of these. After absorption, atoms and molecules become excited. They give out energy quickly either by losing energy in the form of heat or by re-emitting electromagnetic radiation. (2) It is not necessary that the radiation passing through the matter may be absorbed completely. The portion of electromagnetic radiation which passes into matter, instead of being absorbed, may undergo scattering or reflection or may be re-emitted at the same wavelength or a different wavelength. (3) When electromagnetic radiation is neither absorbed nor scattered, it may undergo changes in orientation or polarisation. (4) In some cases, the molecules after absorbing radiation become excited but they do not lose energy very quickly but with some delay. In such cases the energy is re-emitted as radiation, usually of longer wavelength than was originally absorbed. This phenomenon is termed as fluorescence. If in some cases, there is a detectable time delay in re-emission, the phenomenon is termed as phosphorescence. Absorption and Emission Spectra : Spectroscopy is mainly concerned with the interaction of electromagnetic radiation with matter. After interaction, there may occur variation in intensity of electromagnetic radiation with frequency (or wavelength). The instrument which records this variation in intensity of radiation is known as spectrometer. There are two ways in which the interactions are observed, firstly that in which the sample itself emits radiation (Fig. 0.4), and secondly, where the sample absorbs radiations from a continuous source (Fig. 0.4). The spectrum obtained in the former case in known as emission spectrum whereas in the latter case it is termed as absorption spectrum.

Sa~ :

=

,~ ~~ ~

spectrometer:

. . --+ EmiSSion Frequency Spectrum

Excited thermally or Electrically

~ Spectrometer I~ ~'fE

~~

of Continuous Radiation

Sample

.--+

Absorption Frequency Spectrum

Fig. 0.4 : Arrangement for Obtaining Emission and Absorption Spectra

Atomic Absorption of Electromagnetic Radiation : When atoms absorb eiectromagnetic radiation, this results in an increase in their electronic energy. The absorption can only occur if the attacking photon of electromagnetic radiation possesses energy which is equal to the energy difference between two quantised energy levels of the absorbing atom. The increase in the electronic energy results in the jumping of electron from an inner orbit to any outer orbit. When the electron is present in any orbit other than the orbit of least energy for it, the electron, as well as the atom as a whole, are said to be in the excited state. When the excited atom reverts to the ground state, it loses energy in one straight jump or through a

1.10

Concepts in Spectroscopy

5s --4p

. . . . t ..

~==

4s --3p ~-- ·······1···· 3s--

dE 3

2 p -...............•.....

2s - - •• - ••.•••••••• - ••

}~~~~'

1s - - ................................ .

Fig. 0.5 : Energy Level Diagram for Sub·shells in Poly Electrons Atoms.

number of steps in the form of a single quantum or photon of electromagnetic radiation. When it happens for a large number of atoms, the total effect is shown in the light spectrum as a spectral line. This explanation is true for atoms like hydrogen and hydrogen-like atoms. However, in the case of a polyelectron atom, multiplicity of absorptions is possible. The absorbed energy may cause a Is~2s electron transition (~El) which corresponds to X-ray radiations or may cause 2s~2p electron transition (~E2) which corresponds to far ultraviolet radiation or may cause 3d~4p transition (~E3) which requires visible radiation (Fig. 0.5). . The spectra obtained from atoms after absorbing radiation consist of spectral lines of discrete wave lengths extending throughout the spectra and are generally obtained from the light sources like mercury, sodium, neon discharge tube, etc. Some salient features of the line spectra are : (i)

The spectral lines are regularly spaced.

(ii) The wavelengths of the lines emitted are the characteristics of the element under consideration. Line spectrum is independent of the compound in which the element may occur. (iii) When the spectral lines are observed by a high resolving power instrument, the different lines are regularly spaced but they differ in their intensities.

0.6 Molecular Absorption of Electromagnetic Radiation When the molecule absorbs electromagnetic radiation, it gets excited and when it returns to the ground state, it emits bands which are characteristics of the molecule concerned and that is why we call this as molecular spectra also. When each band is observed by a high resolving power apparatus, each band is seen to be composed of a large number of very fine lines. If we observe the lines along the wavelength edge, the lines become closer and closer until they coincide. This edge is known as band head. It has been well-established that spectra of this type are really due to the molecules, which are based on the fact that such bands disappear, when the emitting substance is heated to a temperature at which the molecules disassociate into atoms. The total energy state of a molecule includes-electronic, vibrational, and rotational components. All these energy components are quantised. The rotation energy levels of a molecule are quite closely spaced (Fig. 0.6). Thus, very little energy is required for pure rotational transitions. Such transitions occur in the "far infrared" and "microwave" regions of electromagnetic spectrum. Studies of absorption in this region provide useful information about molecular structure but have found few applications in analytical chemistry. Vibrational levels are farther apart as shown in Fig. 0.6. If the vibrational energy of a molecule is to be increased, more energetic photons are required than the rotational transitions. The vibrational transitions occur in the infrared region of spectrum. It is important to remark here that pure vibrational transitions cannot take place because they are superimposed by rotational transitions. Thus, a typical vibrational absorption spectrum consists of complex bands rather than single lines. When the molecule absorbs photons of electromagnetic radiation in the visible light and ultraviolet radiation, its electronic energy increases, resulting in the electrons to move out to new orbitals of higher energy. A given electronic change is also accompanied by a vibratiomil as well as a rotational change.

Concepts in Spectroscopy

1.11

v.-----------------------------------V.------------------------------------ o

Vs

::1/):

V3----------------------------~------·C

e

'0

V2----------------------------~------ iIi ~

v, ---------------------------~...,.....---- ~ .~

Vo-------------------------r++~~----

W

r V.---____________________

-+++~~----

~

Vs

~

_ V3 __________________

0 ~~~~++~~---·C

e

g

~:==_III(i ~; ~2

Vo

Rotational Vibrational Transitions Rotational Transitions

e

(!)

Electronic Vibrational Rotational Transitions

Fig. 0.6 : Two Electronic Levels are Shown

These changes introduce "fine structure" into the spectrum, so that the molecular spectrum involves a band of wavelengths rather than a single line. Thus, for electronic transitions, a broad absorption spectrum is observed. Emission and Fluorescence If the energy states of the samples are so close to each other and so numerous, as in an excited molecule, that the energies of the various transitions to the ground state cannot be separated, an emission band spectrum is obtained. This principle is applied in molecular fluorescence and phosphorescence instruments. If the emitting energy is quantitatively equal to the excitation energy the process of emission is known as resonance fluorescence. Refraction and Dispersion The refractive index of a medium is the ratio of the velocity of light in vacuum to its velocity in the medium. Since the velocity is dependent on the wavelength of radiation, the refractive index will also be dependent upon the wavelength as well as upon the composition of the matter through which it is passing. The relationship between the refractive index n, the velocity of light in vacuum, c, and the velocity in the medium v is c n=(0.6) v

1.12

Concepts in Spectroscopy

Absorption

+--v

Refractive Index

--*------------------------------_.

-or

~rl

+--v Fig. O.6A : Relation of Refractive Index with Frequency

The change in refractive index which occurs with a change in the wavelength of the transmitted light is known as dispersion. Associated with the absorption of radiation is a related change in the real part of the refractive index. The graphical form of this relation is shown in Fig. 0.6A. It can be seen that the refractive index rises on the low frequency side, reaches a maximum, passes through its mean value at the centre of the absorption, reaches a minimum, and finally approaches a horizontal asymptote on the high frequency side. This asymptote is lower than the high frequency asymptote by a small difference, An, which is proportional to the area of the absorption curve. This is a special case of the Kramers-Kroning relations which connect the absorption at any frequency to the whole of the refractive index curve and thus relate the refractive index at any frequency to the whole of the absorption curve. Scattering and Reflection If the incoming radiant energy strikes upon particles which are suspended in a medium having refractive index different from that of the suspended particles, the light which is transmitted at angles other than 180 0 from the incident light is said to be scattered as the radiation passes through the sample. Nephelometry and turbidimetry are based upon this ability of particles to scatter light. The size, shape and concentration of colloidal particles and suspensions may be determined from this property. Reflection occurs as a ray of light strikes a boundary between two different media. It is dependent upon the refractive indices of the two media and increases as these differences in the refractive indices increase. Selection Rules for Molecular Spectra The molecular spectra are governed by the so-called -selection rules which specify the changes in the quantum numbers accompanying a particular transition. The chemist is lucky that selection rules exist

Concepts in Spectroscopy

1.13

which determine a spectrum. I; there were no selection rules, the resulting spectrum would be very chaotic, indeed! The selection rules are, in fact, the 'backbone' of spectroscopy and are obtained from the quantum theory of interaction of radiation with matter. Let us enumerate a few examples which would later be elucidated. For a diatomic molecule, such as H 2, NO, CO, etc., the selection rule for a pure rotational transition is ~J = ± 1, where J is the rotational quantum number. The selection rule for a pure vibrational transition is ~v = ±l, where v is the vibrational quantum number. The selection rules, however, are not always obeyed strictly. This is because certain approximations which have been used in the derivation of the selection rules are not valid strictly. The spectral transitions which obey a given selection rule are called allowed transitions whereas those which violate a selection rule are ca!led forbidden transitions. In general, the allowed transitions are more intense (stronger) than the forbidden transitions which are weak.

0.7 Types of Molecular Spectra The various types of spectra given by molecular species, the regions in which these spectra lie and the energy that take place in the molecules on absorption of radiation, are given as follows : 1. Rotational (Microwave) Spectra: These spectra arise due to transitions between the rotational energy levels of a gaseous molecule on the absorption of radiations falling in the microwave region. These spectra are shown by molecules which have a permanent dipole moment, e.g., HCI, CO, H20 vapour, NO, etc. Homonuclear diatomic molecules such as H 2, CI 2, etc., and linear polyatomic molecules such as CO 2, which do not have a dipole moment, do not show microwave spectra. Microwave spectra appear in the spectral range of 1-100 cm- I .

2. Vibrational and Vibrational Rotational (Infrared) Spectra : These spectra arise due to transitions induced between the vibrational energy levels of a molecule on the absorption of radiation belonging to the infrared region. IR spectra are shown by molecules when vibrational motion is accompanied by a change in the dipole moment of the molecule. These spectra appear in the spectral range of 50~000 cm- I . 3. Raman Spectra : Raman spectra relate to vibrational and/or rotational transitions in molecules but in a different manner. In this case, only the scattering is measured but not the absorption of radiation. An intense beam of monochromatic radiation in the visible region is allowed to fall on a sample and the intensity of scattered light is observed at right angles to the incident beam. Most of the scattered light is having the same frequency as the incident beam (this is called Rayleigh scattering). However, a small amount of light is having different frequencies than the incident beam. This is known as Raman scattering. The energy differences between these weak lines and the main Rayleigh line correspond to vibrational and/or rotational transitions in the molecule under investigation. Raman spectra occur in the visible region, viz., 12,500-25,000 cm- I . 4. Electronic Spectra : Electronic spectra result from electronic transitions in a molecule by absorption of radiations falling in the visible and ultraviolet regions. While electronic spectra in the visible region span 12,500-25,000 cm- I , those in the ultraviolet region span 25,000-70,000 cm- I . As electronic transitions in a molecule are invariably accompanied by vibrational and rotational transitions, the electronic spectra of molecules are highly complex. There are also the so-called photoelectron spectra (PES) which is used in the determination of ionization energies of molecules. If a light photon falling on a molecule has very high energy, it can bring about ionozation of the molecule, i.e., the removal of the electron from the molecule. If the energy of the incident photon is greater than the ionization energy, the ejected electron will have excess kinetic energy. In PES, a beam of photons of known energy is allowed to fall on the sample and the kinetic energy of the ejected electrons is measured. The difference between the photon energy and the excess kinetic energy is regarded as the measure of the binding energy of the electron. PES is one of the most accurate methods for determining the ionization energies of molecules. Photoelectron spectra can be studied either using the X-ray photons or the UV photons. In the former case, they are called XPES spectra and in the latter case, the UVPES spectra.

1.14

Concepts in Spectroscopy

5. Nuclear Magnetic Resonance (NMR) and Nuclear Quadrupole Resonance (NQR) Spectra. NMR spectra arise due to transitions induced between the nuclear spin energy levels of a molecule in an applied magnetic field. NQR spectra arise due to the transitions between the nuclear spin energy levels of a molecule arising from the interaction of the unsymmetrical charge distribution in nuclei with the electric field gradients (EFG) which arise from the bonding and non-bonding electrons in the molecule. NMR and NQR '>pectra span the radio frequency regions viz., 5-100 MHz. 6. Electron Spin Resonance (ESR). or Electron Paramagnetic Resonance (EPR) Spectra. ESR spectra arise due to the transitions induced between the electron spin energy levels of a molecule in an applied magnetic field. These spectra are exhibited by systems which contain odd (unpaired) electrons such as free radicals and transition metal ions. Molecules such as nitric oxide and oxygen and other paramagnetic systems also show ESR spectra. This branch of spectroscopy is included in the microwave region, viz., 2-9.6 GHz. 7. Mossbauer Spectra (also called Nuclear Gamma Resonance Fluorescence (NRF) Spectra. Mossbauer spectra is a type of nuclear resonance spectra like nuclear magnetic resonance spectra. How ever, while NMR spectra arise due to absorption of low energy photons of frequency around 60 MHz, Mossbauer spectra arise due to absorption of high energy y-photons of frequency around 1013 MHz by the nuclei. Gamma ray spectra are used specifically for the study of compounds of iron and tin. In this case, y-radiations from 57Co source are allowed to fall on a sample in which the iron nuclei are in an environment identical with that of the source atoms. This gives rise to resonant absorption of y-rays. The splittings in Mossbauer lines are of the same order as in NMR spectroscopy. 0.8 Emission of Radiant Energy by Atoms and Molecules (Methods of Electronic Excitation of Atoms and Molecules) Molecules and atoms emit radiation in all regions of the spectrum, from radio waves to radioactive radiation. The most commonly exploited region of the spectrum is the UV range. This range is involved in the excitation of the outer electrons of a molecule or atom, called electronic excitation.

+ hv

Excited Atom

Unexcited Atom

Fig. 0.7 : The Emission 3pectral Energy

When atoms or molecules absorb radiant energy, thereby becoming excited, they usually remain in the excited state for a very short time (in the neighbourhood of 10-6 sec). The atom or molecule then emits a photon of energy and returns to the ground, or unexcited, state. Figure 0.7, which illustrates the process, does not, however, represent the paths of the electrons around the atomic nucleus. These paths are not clear to us at present. We are only able to derive the probability of an electron's being in any one place at any particular time. Nevertheless, this probability does reflect the charge distribution that the electron contributes to the atom as a whole. The energy levels of an atom of one element are different from the energy of an atom of a different element. In all atoms there are numerous energy levels in which the electron may exist. As we saw earlier, the difference in energy between these levels is equal to £2£\ or E3-E2 or £4-£3' and so on (Fig. 0.8). The energy levels are defined by quantum theory. For example, the lowest electron shells of sodium are filled and the valence electron is the 3s electron. All upper orbitals are empty and constitute the upper excited electronic energy levels. In the case of sodium, Eo is the 3s level, £\ is the 3p level, £2 is the

Concepts in Spectroscopy

1.15

Energy change

E3- E2 Fig. 0.8 : Emission of Energy Between Different Excitatior. Levels

4p level, and so forth. All empty upper orbitals are available, but the rules concerning forbidden and permitted transitions must be obeyed. For each change, the energy jump from one level to the next is different, and therefore hv is different. Hence for each transition radiation is emitted at a different frequency v. This means that each element is capable of emitting radiation at many different characteristic wavelengths. These different emission lines constitute the emission spectrum of the element. The emission spectrum of each element is different from that of other elements. This difference provides the basis of emission spectroscopy and enables us to identify the element emitting the radiation. It provides the basis of the most important method of elemental qualitative analysis. For many elements literally hundreds of transitions can be identified from the standard tables. The same type of energy changes take place with electrons in molecules, except that in this case the electrons involved are in molecular orbitals, not atomic orbitals. There are fewer molecular orbitals; hence molecular spectra are comparatively simple. Molecular emission gives rise to molecular fluorescence or phosphorescence. Not all molecules fluoresce, and very few phosphoresce. This phenomenon can be used to detect and identify very small concentrations of certain compounds. The analytical applications of radio frequency radiation and far-infrared radiation are limited but are being studied. Energy changes involving inner shell electrons result in the absorption or emission of x-rays. The most commonly used energy changes are those involving outer shell electrons, such as in emission spectrography, UV absorption, flame photometry, atomic absorption spectroscopy, and so on. The phenomena involving outer shell electrons are described below. 1. Excitation Involving an Electrical Discharge There are several methods of exciting atoms. One of the earliest and most important of these is to put the sample in an electrical discharge between two electrodes (Fig. 0.9). The electrical discharge, of very high energy, breaks down the sample, which is probably in the form of molecules or ions, into excited atoms. Atoms of the different elements present emit a characteristic emission spectrum. This IS the basis of emission spectroscopy. Recently excitation using inductively coupled plasma has been developed and has significantly extended the sensitivity and use of emission spectrography in quantitative and qualitative analysis.

Electrical discharge Sample

Fig. 0.9 : Excitation by Electrical Discharge

1.16

Concepts in Spectroscopy

2. Flame Excitation Atoms may also be excited by putting them into a flame, the thermal energy of which breaks down or reduces the molecules or ions of the sample into atoms. The atoms become excited and emit their characteristic spectra. This is the basis of flame photometry. The energy of a flame is lower than that of an electrical discharge; hence atoms can be excited only to lower states by this method. Fewer transitions are possible; therefore the spectrum consists of fewer spectral lines than in emission spectroscopy. This subject is discu~sed in greater detail later. 3. Excitation by Radiation By irradiating atoms with light of the correct wavelength, can cause the atoms to absorb the radiation and to become excited in the process. The excited atoms reemit the absorbed radiation and become unexcited; the emission of radiation following excitation by radiant energy is called fluorescence. The process of fluorescence of radiation by atoms is the basis of x-ray fluorescence and atomic fluorescence. It is also the basis of molecular fluorescence, a very important analytical field. In practice, the sample is irradiated with UV light, which is absorbed by the molecules. The molecules become excited and reemit the spectral energy. The reemitted fluorescence radiates in all directions. By measuring the intensity of fluorescence at right angles to the radiation that excites the sample, we can distinguish between the beam of light causing excitation and the fluoresced radiation emitted by the sample (Fig. 0.10).

.., Exd'ng "dl.'on :

t

......'Y

I, S'~Ple, 1"~abS~'bed

.... Ii

i

"dla'on

:~..... Fluorescent

1

!

o

".

•

radiation from sample

Detector

Fig. 0.10 : Molecular Fluorescence

Excited molecules normally return to the unexcited state ·without any other change taking place in the molecular orbital. Occasionally, however, the electron involved changes its direction of spin while in the excited state and enters a triplet state. This spin change makes emission of radiation difficult, because the transitions involved are forbidden. In practice, this emission of radiation is delayed compared to fluorescence. This is the basis of phosphorescence, which is not an important analytical field because of the strict controls needed to make the intensity of phosphorescence a reliable measure of the concentrations of emitting molecules present. 0.9 Characteristics of Spectral Lines The natural line width (or life-time broadening) of a spectral line can be determined by the Heisenberg uncertainty principle, Mllt ~ hI41t, where M refers to the uncertainty in the energy and llt is the uncertainty in the life-time of the energy level. Since for a photon E = hv so that I1E = hl1v, hence the natural line width, I1v, given as follows: I1v ~ (41tllttl

(1)

Width and Intensity of Spectral Lines. If we analyse the spectrum of a molecule, the first thing we wish to know is how sharp and how intense (strong) is the spectral line. In other words, it means that

Concepts in Spectroscopy

1.17

Emn Fig. 0.11 (a) : Sharp Spectral Line

Fig. 0.11 (b) : Spectral Line having a width

we are interested in the width and intensity of a spectral line. These two quantities are common to all branches of spectroscopy. Fig. O.l1(a) shows a sharp spectral line having no width while Fig. O.ll(b) shows a spectral line having a width M at half-height. The chemist would, appreciate if the spectral lines were all very sharp and very intense. In practice, this is not so. Two factors which contribute to broadening of a spectral line are : (i) The collision-broadening and (ii) the Ddppler-broadening. The collision-broadening is mainly responsible for the width of spectral lines in the hltraviolet (UV) and visible regions. These transitions mostly occur between electrons in the outer shells in a molecule. When molecules in the gaseous or liquid phase collide with one another, they deform the charge clouds of the outer electrons thereby slightly perturbing the energy levels of these electrons. Hence, the spectral transitions between these perturbed energy levels get broadened. The Doppler-broadening arises if the molecule under investigation is having a velocity relative to the observer or observing instrument. This is generally the case with gaseous samples where the molecules are undergoing random motion according to the postulates of the kinetic theory of gases. If the molecule is moving towards the measuring instrument with velocity u, then the frequency v' of radiation 'seen' by the molecule is given as follows : v' = v (1+ u/c)

(2a)

where v refers to the radiation frequency and c is the velocity of light. If, on the other hand, the molecule is moving away from the measuring instrument, the frequency of radiation 'seen' by the molecule is given by v' = v (i-ule) (2b) Rearranging Eq. 2a, we get (v-vJlv = dvlv = -ule Similarly, rearranging Eq. 2b, we get

(3a)

(v-vJlv = dvlv = ule (3b) The quantity dv is the Doppler broadening. We see that, depending upon the direction of motion of the molecule relative to the instrument, the observed frequency becomes either higher or lower than the actual radiation frequency. From the kinetic theory of gases, it can be shown that the Dopplerbroadening of the spectral line of a molecule of mass m is as follows : dvlv = (21e) (2kT In 21m)1/2 (4)

As dvlv is directly proportional to T1I2, Doppler-broadening can be reduced (and spectral lines of maximum sharpness can be obtained) by working with cold gaseous samples. Let us now turn to the intensity of spectral lines. The intensity of a spectral line can be determined by (i) the Boltzmann population of the energy levels and (ii) the transition probability between the energy levels. According to Boltzmann if, at a temperature T, No is the number of molecules in the ground state, then the number of molecules, N, in the excited s,tate is given as follows : N = No e-MlkT (5)

1.18

Concepts in Spectroscopy

where AE refers to the energy difference between the ground and excited states and k is the Boltzmann constant. The relative population at equilibrium is, thus, given as follows :

NINo

=

e-tlElkT

(6)

Evidently, if AE is large, NINo is small, i.e., the number of molecules in the excited state is less than in the ground state. In fact, at room temperature, most of the molecules are in the ground state. Hence, the spectral lines originating from transitions from the ground state to a higher, say, third excited state would be more intense than those originating from transitions from the first excited state to the third excited state. The term transition probability means the probability of transition between two given energy levels. It is known that a molecule, upon absorption of a photon, does not just go anywhere it pleases ; it ends up in an energy level determined by the selection rule for the particular transition. The selection rules, in. tum, ascertain the allowed transitions and the forbidden transitions. Clearly, the allowed transitions yield spectral lines which are having greater intensity than the forbidden ones.

Complexity of Spectra and Intensity of Spectral Li!les The number of intensities of lines which may appear in a spectrum can be determined by three factors. 1. the populations of the energy levels from which the transitions originate 2.

the values of individual transition moments or transition probabilities

3.

quantum mechanical selection rules.

In brief, the statistical probability for the occurrence of a transition can be calculated and lies between zero and one. It can be shown that weak spectral lines have small transition moments while strong lines have values approaching one. One factor which determines the value of a transition moment is the magni-tude of the change in dipole moment associated with the transition. In addition, the selection rules may show that a particular transition is forbidden. As a consequence of these conditions, spectra often have fewer lines or bands than might be expected, a fact which facilitates interpretation. A d:!tailed discussion of transition moments and selection rules, which are mathematical concepts, is beyond the scope of this book and the known selection rules will therefore be assumed. Spectral line widths also affect the appearance of the spectrum. Lines arising from transitions in atoms and gaseous molecules are characteris-tically narrow as are those due to electron and nuclear spin transitionS which involve only small energy changes. Molecular absorption spectra in the infrared, visible and ultraviolet regions consist of sets of very closely spaced lines which are not normally resolved by the instrumentation used. They are broadened by collisions between solute and solvent molecules, so that the overall appearance is of a number of broad overlapping band envelopes. The relative populations of energy levels, that is the proportion of the analyte species occupying them, have a direct bearing in line intensities and are determined by the spacing of the levels and the thermodynamic temperature. The relation is expressed in the Maxwell..Boltzmann equation,

(i) where n 1 and n z are the numbers of species in energy states £1 and £z separated by AE, gl and gz are statistical weighting factors, R is the Boltzmann constant (1.38 x W-Z3 J K-I) and T is the thermodynamic temperature. Calculations show that at room temperature and when AE exceeds 103 J mol-l, only the lowest level of a set will be populated to a significant extent. Thus, absorption spectra recorded in the infrared, visible, ultraviolet regions and beyond arise from transitions from the ground state only. Furthermore for atomic emission spectra to be observed, a considerable increase in temperature (to more than 1500 K) is required to give appreciable population of the higher energy levels. Indeed the practical importance of the Maxwell-Boltzmann equation lies in demonstrating the effect of changes in thermal excitation conditions on the intensity of atomic emission spectra. Therefore it is possible to have some measure of control over the sensitivity of atomic emission techniques but little or no such control over molecular absorption.

Concepts in Spectroscopy

1.19

0.10 Theoretical Principles of Atomic Spectroscopy Introduction : When a beam of polychromatic light is passed through a prism or grating, it is broken up into its constituent colours. This array of colours is known as Spectrum. The spectrum what we see in only the visible region which extents from 4300 to 6900 A 0. However, the complete spectrum may extend over very wide range from y-rays of wave lengths 10- 13 meter to radio waves of wave length 105 meters. There are two principal classes of spectrum, known as emission spectrum and absorption spectrum. Let us discuss these one by one. 1. Emission Spectrum : This type of spectrum may be obtained, when the light coming after passing through a prism or a grating, is examined directly with a spectroscope. Emission spectrum is further classified according to their appearance as continuous, line and band spectrum. (a) Continuous Spectrum: When the source emitting light is an incandescent solid, liquid or gas at a high temperature, the spectrum so obtained is continuous. In other words, this type of spectrum is obtained whenever matter in the bulk is heated. For example, hot filament, hot iron, hot charcoal give the continuous spectrum. However, some of the important characteristics of continuous spectrum are : (i) It consists of a wide range of continuous wavelength lines from red to violet which appear as confuuous luminous band of light. (ii) The intensity of spectrum is not uniform over the entire spectrum. It becomes maximum at a particular wavelength and decreases on either sides. It is important to remark here that the point of maximum intensity shifts towards the violet when the temperature of light emitting solid is raised. (iii) The general appearances of the continuous spectrum are independent of the nature of the light emitting substance. .

(b) Line Spectrum : This is obtained when the light emitting substance is in the atomic state. Therefore, it is also called atomic spectrum. Line spectrum consists of discrete wave-lengths extended throughout the spectrum and is generally obtained from the light sources like mercury, sodium, neon discharge tube, etc. However, some salient features of the line spectrum are: (i) In this, the spectral lines are regularly spaced. (ii) The wave-length of the lines emitted are the characteristics of the element under consideration. Line spectrum is independent of the compound in which the element may occur. (iii) When these lines are observed under a high resolving regularly spaced but they differ in their intensities.

po~er

instrument, the different lines are

(iv) Line spectrum can be observed against a dark or a faint continuous background. (c) Band Spectrum : This type of spectrum arises when the emitter in the molecular state is excited. Each molecule emits bands which are characteristics of the molecule concerned and that is why we call this as molecular spectrum also. The sources of band spectrum are (i) carbon arc with a metallic salt in its core (ii) vacuum tubes, etc. However, the following are the characteristics of band spectrum. (i) It consists of luminous bands separated by dark spaces. (ii) When each band is observed under a high resolving power apparatus, each band is seen to be composed of a large number of very fine lines. If we observe the lines along the long wave-length edge, the lines become closer and closer until they coincide. This edge is known as 'band-head'. (iii) The lines of band constitute characteristics of the molecules and the method of their excitation. 2. Absorption spectrum : When the light from a source emitting a continuous spectrum is first passed through an absorbing substance, and then observed through a spectroscope, it will be found that certain colours (or wave lengths) are missing which leave dark lines or bands at their places. The observed spectrum will be called absorption spectrum. The region of spectrum absorbed depends upon the nature of the absorbing material. Like emission spectrum, absorption spectrum is ;}lso of three types:

1.20

Concepts in Spectroscopy

(a) Continuou$ absorption spectrum: This type of spectrum arises when the absorbing material absorbs a continuous range of wave length. An interesting example is one in which red glass aJ->sorbs all colours except red and, hence, a continuous absorption spectrum will be obtained. (b) Line absorption spectrum: In this type, sharp dark lines will be observed when the :tbsorbing substance is a vapour or a gas. The spectrum obtained from sun gives Fraunhofer absorption lines corresponding to vapours of different elements which are supposed to be present on the surface of sun. (c) Band absorption spectrum: When the absorption spectrum is in the form of dark bands, this is known as band absorption spectrum. An interesting example is one when an aqueous solution of KMnO 4 gives five absorption bands in the green region. Spectroscopic Displacement Law W. Kossel and A. Sommerfeld in 1919 gave the spectroscopic law according to which, the arc spectrum of an element is similar to the first spark spectrum of the element one place higher in the periodic table or to the second spark spectrum of the element two places higher in the periodic table and so on. The law is generally applicable to fine structures of the spectral lines. Different Spectral Lines of Hydrogen Every atom when excited emits radiations. The radiations form a line spectrum which is the characteristic of the emitter. Each atom has its own particular line spectrum which is regarded as the characteristic of that element to which the atom belongs. Like other elements, hydrogen possesses its, own characteristic line spectrum. Hydrogen spectrum consists of a number of lines. These have been grouped into five series which are named after their discoverers, (Fig. 0.12). Many attempts were made by various workers to find a rule or underlying relationship which governed be wave lengths of these lines. We will discuss these one. BALMER

LYMAN

I111 I 1- 1111111 Fig. 0.12: Hydrogen Spectrum

(a) Balmer's Law: The actual development of the spectroscopy started in 1885 when J.J. Balmer found that the wave-length of the hydrogen lines could be represented by the formula :

A. = B (

2n2

n -2

2)

(0.7)

where B is a constant, n is an integer with variable values 3, 4, 5, ... Eq. (7) may be written as : (0.8)

or

V=~=~(_1 . A.

B 22

__n1)=R(_122 __n1) 2

2

(0.9)

where y, the reciprocal of wavelength, is called wave number. Its unit is cm- 1 and, it represents the number of wave lengths per crn. The value of R was found to be 109,677.6 cm- 1 for the visible lines of hydrogen. Rydberg showed that this constant R is not the property of only hydrogen spectrum but it must occur in the atomic spectra of all elements. There is a close agreement between the calculated and observed values of the wavelengths of spectral series of hydrogen for various values of n. This is shown in Table 0.2.

1.21

Concepts in Spectroscopy

It can be seen that the lines become close together as the value of n increases. These lines converge at v = RI4 cm- i or A. = 3647.45Ao when n approaches infinity. This value of A. at which the series converges is called convergent limit. After the discovery of the Balmer's series, attempts were made to discover the other lines. Besides this Balmer's series in the visible part, Lyman found a series in ultraviolet region. The wave-length of lines of the Lyman series is given by :

~ = R C12 - n12 )

where n = 2, 3, 4 ........ .

(0.10)

Table 0.2 : Spectral Lines of Hydrogen Value of n

Observed AO

Calculated A 0

3

6562.80

+0.01

4

6562.79 4861.33

4861.38

+0.05

5

4340.47

4340.51

+0.04

6

4101.74

4101.78

+0.04

7

3970.06

3970.11

+0.05

8

3889.00

3889.09

+0.09

Difference A 0

(b) Rydberg's Formula: Rydberg, as a result of detailed researches, established in 1889 that all the series in optical spectrum could be arranged accordmg to a general relation' of the form

R

v = voo - ---,;(n+b)2

(0.11)

where R was found to be a universal constant for all series, now called the Rydberg constant, n an integer, b a fraction less than unity which is practically constant for all the lines of a series, ~co the limiting or convergent wave number in the series corresponding to n = 00. He proved also that the Balmer's formula is a special case of the above general relation [Eq. (0.11 )]. The great merit of Rydberg's formula consists of the fact that wave number of any line can be expressed as the difference of two terms, one fixed, represented by m and the other variable which is obtained by giving different integral values to n. Rydberg's formula can therefore be written in a simple form as (0.12) where m is fixed and n is variable. Following up the work of Liveing and Dewar, Rydberg was able to identify the four different sets of spectral lines in alkali spectra. All of these follow the equation (0.13).

R

R

(m+a)2

(a+b)2

v=-----=-

(0.13)

Four of these series are of particular interest, mainly due to the continued used of the symbols S, P, D and F for sharp, principal, diffused and fundamental series of spectral lines respectively. These are

R

(i)

Principal series,

v=--:=-

(ii)

Sharp series,

v=---:::-

(l+sf R

(2+p)2

R

---:=-2 '

(n+P)

R

--'""'2:.-'

(n+S)

n = 2, 3, 4, ...

n = 2, 3, 4, ...

1.22

Concepts in Spectroscopy R

V= ----,::-

. (iii) Diffuse series,

(2+p)2

R (n+D)

---2=-' n = 3, 4, 5, ...

R

R

--~2'

v=--~

(iv) Fundamental series,

{3+D)2

n=4, 5, 6, ...

(n+F)

Here S, P, D and F are constants that are characteristics of the particular series. (c) Ritz Combination Principle: It occurred to Rydberg that combinations of terms other than those giving the four chief series might correspond to new lines observed to be present in spectrum but belonging to the new series. Ritz, in 1908, generalised this idea of Rydberg into a principle which achieved remarkable results in the classification of spectral lines. It may be stated thus : "By a combination of the terms that occur in the Rydberg or Balmer formula, other relations can be obtained which hold good for new and new series. " For instance, series, other than of Balmer in the hydrogen spectrum were predicted even before they were actually discovered by Paschen and Brackett. Taking the first two lines, Ha and H{3 of the Balmer series, we may represent them by

- R[I22-32"I ]and - = R[122-42"l"J va =

v~

Combining these two equations as (0.14) The above expression represents a new line indeed, the first line of a new series in the infra-red discovered by Paschen. Similarly, the second line of the same series can be obtained by finding the difference between Hyand Ha and so on. In like manner, another series, also in the infra-red, discovered by Brackett, can be obtained with the combination principle. Ritz's principle gives the clue to interpret atomic spectra in terms of quantum theory. The Spectrum of Singly Ionized Helium '\ In case of singly ionized helium, four spectral lines were detected. He+ contains only one extranuclear electron, and therefore its spectrum showed similarity to that of hydrogen. The different series can be expressed by the following general formula, i.e., (0.15)

where Z is known as effective nuclear charge having value 1 for neutral atom, 2 for a singly ionised atom, 3 for a doubly ionized atom and so on. The four series correspond to n 1 values of 1, 2, 3 and 4 respectively and n 2 = n 1+l, n 1+2, n 1+3 and so on. Spectra of Higher Elements The spectra of the higher elements particularly those belonging to the first three groups of the periodic table, were found to contain four different spectral lines. These are called sharp, principal, diffused and fundamental and are denoted by S, P, D and F respectively. The wave-length of the each serir.s is given by the following expression : (0.16)

Concepts in Spectroscopy

1.23

where L gives the frequency of the convergence limit of particular spectral series and 0 is a quantity approximately constant for any given series. For alkali metals, 0 is actually constant. Explanation of Origin of Atomic Spectra Following are the attempts to explain the origin of atomic spectra. I. Rutherford's Explanation : According to Rutherford, an atom consists of central nucleus surrounded by revolving electrons. Objection : If the above model of Rutherford is accepted, then according to Maxwell, an electron should continuously lose energy in the form of radiations and come closer to the nucleus by following a spiral path until it ultimately falls into it. If the Maxwell's objection is accepted, the spectral lines produced by a moving electron would drift towards the violet end of the spectrum but elements ha\ e found to emit spectral lines of fixed frequencies like the Balmer series of hydrogen. II. Bohr's Theory: Niel Bohr applied the concept of Max Planck (Quantum theory) to the problem of revolving electron around hydrogen nucleus and gave the following main points of what is now known as Bohr's theory of atomic structure. First Postulate : That within the atom an electron can move in certain specific orbits without radiating out any energy. This orbital rotation without emitting energy (radiatIOn) follows the Newtonian's Law, i.e., the force of attraction between the nucleus and electron is equal to the centrifugal force of the moving electron. Second Postulate : Bohr postulated that the electron moves in certain discrete non-radiating orbits, called stationary orbits for which the total angular momentum of the moving electron is an integral multiple of hI21t, where 'h' is called Planck's constant. h mur=n(0.17) 27t Here n is called the principal quantum number and is equal to 1, 2, 3, 4 ... etc. This equation is called quantum condition and limits the number of permissible orbits. Third Postulate : Bohr assumed that no energy was radiated out by the atom, so long as the electron continues to move in its stationary orbits. The energy is only radiated out when an electron jumps from one stationary orbit to another. The frequency of the emitted radiation (v) is given by the difference in energy between the initial and final orbits. E 1-E2 = hv (0.18) This equation is known as Einstein's frequency condition. Let us now apply the above conditions to a hydrogen atom.

Fig. 0.13 : Motion of an Electron In an Orbit

Concepts in Spectroscopy

1.24

Radii of Electronic Orbits or Shells: Consider a nucleus with the positive charge '+Ze' and an electron with negative charge '-e' revolving around it at a distance r (Fig. 0.13). Let u be the velocity of the electron. There are now two forces acting on the electron which keep it revolving around the nucleus: (a) The centrifugal force equal to mu 2/r acting away from the nucleus and (b) The electrostatic force of attraction equal to Ze2/r2 acting towards the nucleus. Since the electron keeps on revolving at a fixed distance r from the nucleus (first postulate), the two forces must balance each other.

mu 2 e 2 Z -r-=7

(0.19)

But angular momentum, p = Iw, where I is the moment of inertia and w, the angular velocity. But

u r

1= mu2, and w=2 u mur2 p= Iw=mr x-=--=mur r r

According to the second postulate the angular momentum of an electron is an integral multiple of hI21t, i. e.,

nh mur =-,n =1,2,3, ... 21t nh u=-21tmr Substituting the value of u from equation (0.21) in (0.19) we get

or

r=

e 2Z(21tmr)2 n 2h 2 or r = --::---::-2 2 mn h 41t 2me2Z

(0.20) (0.21)

(0.22)

This gives the radii of the stationary orbits by substituting n = 1, 2, 3, ....... and so on. If n = 1, h = 6.63xlO-27 erg-sec, Z = 1, m = 9.1xl0-28 gm., and e = 4.8xlo--lO e. s. u., we get r = 0.52xl0-9 em. This is the smallest possible radius of the orbit of the electron in the hydrogen atom. (For hydrogen atom Z = 1). (a) Energy of the revolving electron: The total energy of the electron at any instant is equal to the sum of its kinetic and potential energies. (i) Kinetic energy : If m is the mass of electron and u is velocity, then K.E. = 'hmu 2. (ii) Potential energy : Potential energy of the electron at a distance 'r' from the nucleus is defined as the work done to bring the electron from infinity to the distance 'r'. If '+Ze' is the charge on the

-e 2 Z nucleus and -e, that of an electron, the work done is given by - r .. P.E.

-e2 Z r

=--

1.25

Concepts in Spectroscopy

Therefore, total energy of the electron is given by the sum of P.E. and K.E., i.e., 2

) 2 e Z E=zmu --r-

From equation (0.19), we have r

(0.23)

e2Z

= mu 2

?

or u-

=

p2 Z

mr

Substituting the value of u2 in equation (0.23), we obtain 2

2

2

E _ ) me Z e Z _ e Z - z-;;;;---r- --2,:

(0.24)

Substituting the value of r from equation (0.22) in the above equation, we get E=-

27t 2 me 4 1Z2 2

(0.25)

'-2-

h

n 1

Substituting the values of m, e, hand n = 1, Z = 1 i.n eq. 0.25 we get E) = -2.18 x 10- 11 erg.

(0.26)

E) also denotes the energy of the unexcited hydrogen atom. E, E) E 2, ..•.••••.. have a negative value because the reference level of P.E. is taken when the electron is at in infinite distance from the nucleus. (For H atom, Z = 1).

Negative Energy of the Electron: The negative sign in equation (0.25) means that the electron is bound to the nucleus by attractive force so that energy must be supplied to the electron in order to separate it completely from the nucleus. As 'n' increases, the absolute numerical value of the energy decreases, but on account of the negative sign, the actual energy will increase. This means that the outer orbits have greater energies than the inner ones. The negative energies of the electron in the orbits appear to be strange but can be explained as follows: When the electron and the nucleus are infinitely far apart, (i.e., ionised atom), a state of zero energy (E = 0 and n = 00) is supposed to exist. Therefore, they do not interact in any way and as such is at rest. As they move close together, they are attracted and energy is released and so the energy of the system becomes less than zero, i.e., negative. Thus, the energy of the electron in an atom is negative as compared to the energy of free electron. The innermost stationary state or orbit (n = 1) has the most negative value and this state is called the ground state. Calculation of Rydberg's constants : The energy of the hydrogen atom when the electron is in the n )Ih orbit : E n)

_ -

2

27t me 2 h

4

1 Z2 '-2 n)

and the energy of the atom when the electron is in the nzth orbit 27t 2 me 4 h2

-;;r 2

and the frequency of the photon emitted, when an electron jumps form postulate, i.e.,

or

n) th

orbit is given by Bohr's third

1.26

Concepts in Spectroscopy

Since the veloci!y of light c

vA, we have

=

1

v c

-=-

or

or

A

..!.. = A

.

2 (_I___I_}Z2

21t me ch 3

n2 2

n 12

..!.. = R(_I___I_}Z2 A

n2 2

2

where

4

R = 21t me ch 3

n12

(0.27)

4

On substituting the values, we get R = 109737.309 cm- I . This value is found to be in agreement with the value obtained from the spectroscopic data of the Balmer's series which is 109677.567 cm- I . Successes of Bohr's Theory: This theory explains the following facts about atomic spectra. (i) Explanation of Emission Spectrum : When an element is heated or supplied with energy by some method, its atoms get excited to higher energy levels, i.e., the electrons junlP from inner to outer orbits. The electron, excited to a higher energy orbit, will immediately try to return to one of the lower energy vacant orbits. An excited electron could do so in one big jump giving a large value of !1E or it could return in stages through orbits of immediate energy values. Each step or stage on the return jump will be associated with the emission of energy in the form of spectral lines. This types of spectrum is called the emission spectrum of the light emitting element. In the emission spectrum, the spectral lines are observed as bright lines against a dark background. (ii) Explanation of Absorption Spectrum : When energy falls on a nomml lower exited state of an atom, the electrons jumps from an inner to an outer orbit, resulting in the absorption of definite quantities of energy. If this excitation energy is obtained from a beam of light, the lines with corresponding frequencies would be missing from the spectrum of light used. In fact, lines of darkness will be seen against background of light in such a spectrum. These dark lines against background of light are termed as the absorption spectrum of light absorbing element. (iii) Explanation of Frequencies of Various Spectral Lines : It explained the frequencies of various lines of hydrogen spectrum. It was explained by Bohr that Balmer lines are produced when the electron jumps from the higher orbits 3, 4, 5, 6, ............ etc., to the lower orbit n = 2 and equation (0.27) gives this value by n2 = 2 and n l is equal to an integer higher than 2.

Balmer's series n 1 = 3, 4, 5, ................. .

where

Bohr predicted the presence of another series which comes as a result of jump of the electron from a higher orbit to the orbit n2 = 1. This is given by Lyman series

..!.. = R(~-~) where nl =2,3,4, .... A 1 n 1

This series was latter photographed by T. Lyman of the Harvard University in 1916. Similarly, he explained the Paschen's series for n2 = 3 and n l = 4, 5, 6, ... and also predicted the presence of other series like Brackett and Pfund series which 'were discovered by other scientists and named after them.

Concepts in Spectroscopy

1.27

Brackett series n 1 = 5,6,7, .......

where Pfund series

where n 1 = 6,7,8, ...... . In Fig. (0.14) a number of closely spaced lines is shown for n is equal to infmity. This corresponds to the fact that electrons knocked out of hydrogen and have kinetic energies representing the difference between the ionisation energy and energies of identical electrons. These electrons outside the hydrogen atoms are not subject to quantum conditions and kinetic energies of these electrons can have any value. The transitions. from continum to any of the levels will give rise to radiations of all frequencies. A continuous spectrum has been observed at the end of Balmer's series.

I

!! Brackett

1

'

...

f'fund

!=;AriAl':

n=oo n=6 n=5 n=4

Series

Paschen Series

n=3

Balmer Series

>-

~

Q)

c:

W

Lyman Series

--------Wave Length---------. .. Fig. 0.14 : Origin of Spectral Lines

(iv) Intensities of spectral lines : When a single electron of an atom jumps, the energy released is so small that the corresponding spectral line will not be detected. The intensity of a line increases if

1.28

Concepts in Spectroscopy

the same electron jumps simultaneously in a number of atoms. As the number of such atoms becomes larger, the more intense spectral lines will be obtained which correspond to more easily possible transitions.

Failures of Bohr's Theory (i) It failed to explain spectra of atoms other than hydrogen. (ii) When spectral lines of hydrogen are observed very closely, each line is further made up of much closely spaced lines. (iii) It failed to explain Zeeman effect. When a substance emitting a line spectrum is placed in a magnetic field, its lines would split up into a number of closely spaced lines. This is known as Zeeman effect (Fig. 0.15). No Field

~~

A~ Weak Field

Fig. 0.15 : Zeeman Effect

(iv) It failed to explain Stark effect. It is similar to Zeeman effect and is produced in the presence of external electrostatic field. (v) Bohr assumed that the nucleus is stationary and only electrons are revolving around it. Detailed facts have revealed that both the nucleus and the electrons move in closed orbits around their centre of mass. Therefore, it is a major weakness of Bohr's theory. (vi) Another weakness of Bohr's theory is that it did not throw light on the distribution and arrangement of electrons in atoms. (vii) In the light of the Heisenberg's uncertainty principle, both the velocity and the position of the electron cannot be specified at a given time as Bohr did. Thus, it is another weakness of Bohr's theory. (viii) Another fundamental objection against Bohr's theory is that it uses two theories which are opposed to each other, i.e., quantum theory was used to account for the existence of stationary orbits and for frequencies of radiations emitted while motion of electrons in its orbit obeyed the law of classical mechanics.

Effect of Nuclear Motion on the Energy Levels So far we have assumed that only the electron is moving. It is known that both the electron and nucleus should move in a force field. Moreover the two particles should move about their common centre of mass, the nucleus being always on opposite side to that of the electron (Fig. 0.16).

Fig. 0.16: Nuclear Motion

1.29

Concepts in Spectroscopy

This is equivalent to the electron having a mass J1 rotating about a stationary nucleus, it is called reduced mass of the electron and is given by the following equation : mM 11=-m+M

(0.28)

where M and m are the actual masses of the nucleus and the electron respectively. Thus the expression for the energy of the nth levels of the electron in a hydrogen-like atom should be given by:

En =

21[2 Z2 e 4

1l

(0.29)

n2 h 2

Now for hydrogen if m = 1 and M = 1836, then

1:. = ~ = 1836 = 0.99945

(0.30) m m+M 1837 Thus, the magnitude of the energy of the electron in all the levels should decrease by a factor 0.99945. The Sommerfeld Model Inspite of many successes, Bohr's theory was found to be inadequate to explain certain details in the spectrum of hydrogen. For instance, the Ha. line of the Balmer series was found to contain several components. This fine SITUcture of spectral lines could not be explained by Bohr's theory which assumed that there was only one orbit for each quantum number n, whereas the observed fme structure suggested that for any given quantum number n, there might be several orbits of slightly different energies. Sommerfeld, in 1915, guided by the above suggestion modified Bohr's theory by introducing the following modification :

o

He (2 Electrons) Li (3 Electrons)

Elliptical Orbit

. Ne (10 Electrons)

Fig. 0.17 : Sommerfeld Elliptical Orbits

1.30

Concepts in Spectroscopy

Concept of elliptical orbits, and (b) Relativistic variation of the mass of electron. We will discuss these modifications one by one. (a) Elliptical Orbits: In order to explain the multiplicity of spectral lines, sommerfeld introduced the concept of elliptical orbits. He postulated that: (i) Since the electron is moving around and under the influence of a massive nucleus, like a planet around the central massive sun, it might describe elliptical orbits as well. (ii) While retaining the first circular orbit suggested by Bohr, Sommerfeld assumed one additional elliptical orbit in the case of Bohr's second orbit and added two additional elliptical orbits to Bohr's third orbit and so on [Fig. (0.17)]. (iii) The nucleus is one of the foci for all these orbits. (iv) In an elliptical orbit, the major and minor axes will differ in lengths, but as the orbit broadens, they will approach each other and become equal when the orbit becomes circular. Circular orbit is only a special case of the elliptical orbit. The orbit is defined by two quantum numbers, i.e., nand k which correspond to the major and minor axes of the ellipse. They are related as follows: (a)

Principal quantum number n length of major axis =-= Azimuthal quantum number k length of minor axis It is clear from the above that for any given value of n, k cannot be zero as in that case the ellipse would degenerate into a straight line passing through the nucleus. Further, k cannot be more than n since b is always less than a. When n = k the path becomes circular. There is a limit to the number of different orbits that an electron may have in any energy level. Such possible paths are referred to as sub-levels. For a given value of n, k can have only n elliptical orbits or sub-levels with different eccentricity. For elliptical orbits the value of k would be (n-l), (n-2) etc. down to k = 1. When k = 0, the ellipse would be a straight line. When n = 3, k = 3 (circular), 2, 1 (two elliptical). Thus, for n = 3, there are three different orbits. Thus the greatest contribution of Sommerfeld model lies in its sub--division of the original Bohr stationary levels into various sub-levels of slightly differing energies as given by differences in orbit shapes.

Fig. 0.18: Motion of an Electron In an Orbit

Simple mathematical approach to Sommerfeld model : Let us consider the motion of an electron (-e) in an elliptical orbit as shown in Fig. 0.18. Its position at any instant can be fixed in terms of polar co-ordinates, rand I/J where r is the distance of the electron from the nucleus (+e) at one of the foci of

the ellipse and I/J is the angle which the radius vector makes with the major axis of the ellipse. The tangential velocity v of the moving electron at any instant can be resolved into two components : (i) One radial, i.e., drldt along the radius vector-Corresponding to this there will be a radial momentum Pr equal to m(drldt).

Concepts in Spectroscopy

1.31

(ii) Other transverse, i.e., at right angles to the radius vector equal to r (dqJ/dt). Corresponding to this

there is angular or azimuthal momentum p 41 equal to mr2

~~ ,

where m is the mass of the electron.

As Sommerfeld considered the circular orbits to be special cases of elliptical orbits, he assumed that the elliptical orbits should satisfy the quantum condition of Bohr just as the circular orbits, i.e.,

nqY

f

Pr dr = nrh

(0.31)

fp'l'd~ = n'l'h

(0.32)

Thus, the single n of Bohr's theory has been replaced by the two new quantum members nr and i.e., n = nr

+ nf{J

(0.33)

where nr is the radial quantum number and nf{J is the angular or azimuthal quantum number. The total energy E is given by E = P.E. + K.E. = P.E. + Radial K.E. + Angular K.E.

=_~+!m(dr)2 +!mr2(d~) r

2

2

dt

dt

(0.34)

From equations (0.3 1), (0.32), (0.33) and (0.34), it can be shown that

• (0.35) where

£

is the eccentricity of the ellipse whose semi-major and semiminor axes are a and b respectively.

and

(0.36)

From equation (0.35), we get

(1- &2 ) 1/2 = ~ = a

n'l' n'l'

+nr

n

Thus, equation (0.36) can be written as E=

27t 2 mZ 2 e 4

--'-~-=--2

n h4

(0.37)

From equation (0.37), it follows that (i) The elliptical orbits which have the same value of n, though of different eccentricities, have same energy. This it is not correct. (ii) All orbits having the same values, of the semi-major axis possess the same energy, since the length for the semi-major axis is detennined solely by the total quantum number n (= nf{J+n,). Hence the energy of any of these permitted elliptical orbits is identical with that of a circl'lar Bohr orbit, whose radius is equal to the semi-major axis of the ellipse. From above it follows that the theory of elliptical orbits inspite of two new quantising conditions involved, introduces no new energy levels other than those given by Bohr's theory of circular orbits. No

1.32

Concepts in Spectroscopy

new spectral lines, which would explain the fine structure, are therefore predicted. Thus, Sommerfeld has to modify his own model by suggesting the variation of mass of electron with velocity relativistic variation.

Fig. 0.19 : Sommerfeld Elliptical Orbit Precessing about an Axis through one of its Foci

.(b) Relativistic Variation of the Mass of Electron : The velocity of an electron moving in an elliptical orbit varies considerably at different parts of the orbit, being a maximum when the electron is nearest to the nucleus. When allowance is made for the relativistic variation of electronic mass, as Sommerfeld did, the path of the electron is found to be no longer a simple ellipse; it is indeed, no longer a closed figure, but is transformed into a complicated curve known as rosette-a processing ellipse (Fig. 0.19). The total energy E of the system corrected by the relativistic variation of the mass of electron can be shown to be

(0.38) The relativity correction, therefore, results in splitting up a given energy level En into n levels differing slightly from one another in energy. The splitting up of each energy levels gives rise to a fme structure of single spectral line, on application of the usual Bohr frequency condition. When one explains the, fine, structure of lines, one should not take into account all theoretical possible transitions of the electron from one elliptical orbit to another that actually occur. According to a principle known as the selection rule, transition can take place only between orbits for which the azimuthal quantum number changes by+l or-I, i.e., nl/J = ±1. Limitations of Sommerfeld Model: (i) Though Sommerfeld's theory is fairly well verified yet it leads only to three components for the structure of Ha line, while there should be really five. Thus the relativistic atom model has met with a partial success. (ii) Sommerfeld model could not explain Zeeman and Stark effect. (iii) It provides inaccurate values for angular momentum. (iv) The model gives no information regarding the relative intensities of lines, whose frequencies alone are predicted. (v) It provides no idea about the number of electrons which can be accommodated in a particular orbit. (vi)

The concept of elliptical orbits due to Sommerfeld gives the correct total (n) of possible azimuthal quantum number, but the actual values are not correct. The experimental studies as well as theoretical treatment based on wave mechanics show that azimuthal quantum number can be zero, so that the values can be 0, 1, 2 .............. n-l, thus making a total of n possibilities. The correct and new azimuthal quantum number is denoted by 'I' to avoid the confusion. Thus, [ is equal to K-l.

Concepts in Spectroscopy

1.33

n,n

Va

· ·· ··

Vb

Ve !

~

+

3.3 3.2 3.1

Vd

Ve

IIj

·· ·· · .L.·

2.2

2.1

Fig. 0.20 : Fine Details of the Line Spectrum

Explanation of Fine Details in Line Spectrum by Sommerfeld Model : It can be explained as follows: When an electron moves in an elliptical path, there will be a displacement each time in its elliptical path and thus resulting in a small difference in energy. Thus, the path of the moving electron is no longer a simple closed ellipse but a complicated curve, known as rosette, which is made up of different elliptical orbits of slightly different energies. This difference in energies of elliptical orbits explains the existence of components of spectral lines in the spectrum of higher elements. Criticism of the theory : (i) Suppose there are two orbits with principal quantum numbers, n = 2 and n = 3 (Fig. 0.20) respectively. Consider an electron which falls from third to the second orbit. There may be six possible transitions, each giving one fine line. But actual observations yield only five lines (Fig. 0.20). Possibilities i.e., 11K = 3-2 = + 1 (i) 3K3~2K2 i.e., 11K = 3-1 = + 2 (ii) 3K3~2Kl (iii)

11K = 2-2 = 0 i.e., 11K = 2-1 = + 1 (iv) 3K2~2Kl (v) 3Kl~2K2 i.e., 11K = 1-2 = - 1 i.e., (vi) 3Kl~2Kl 11K = 1-1 = 0 where 11K indicates the change in azimuthal quantum number. Sommerfeld applied selection rule which limited the number of transitions. According to such rule only those transitions are possible for wl).ich the quantum number K changes by-l or+l, i.e., 11K = +0. Hence transitions like (3Kl~2Kl)' (3K3~2Kl) and (3K2~2K2) are forbidden. Therefore, according to this theory, the fine structure of Hu lines should made only of three lines. But actually it splits up into five lines. Sommerfeld theory fails to explain complicated systems. (ii) Sommerfeld's model could not explain Zeeman and Stark effect. (iii) The model gives no information regarding the relative intensities of lines, whose frequencies alone are predicted. IV. Wave Mechanics and Spectral Lines: According to wave mechanics orbits do not exist in the atom. Thus, the interpretation of the emission of radiation due to a jump of the electron from outer to the inner orbits does not hold good. In wave mechanics, place of orbits, have been taken by stationary states of atoms with definite energies. When any state excited by the absorption of energy comes to the ground state, spectral lines 3K2~2K2

i.e.,

1.34

Concepts in Spectroscopy

are produced. Thus, the frequency of each spectral line may be regarded as a 'beat' frequency between two states of the atom, which gives the same result as that of Bohr. As '112 represents the electrical charge density, then in the stationary state this remains constant and so also the charge density. But when any state becomes excited by the absorption of energy, 'II is no longer constant and varies periodically and so also the charge density. This periodic variation in charge density is accompanied by the emission of radiation. When an atom absorbs energy so that its two stationary states become excited simultaneously, two superimposed sets of radiations are produced to give a 'beat' variation. If v' and v" represent the frequencies of two superimposed vibrations, the frequency v of the emitted beat is v = v' - v" The wave mechanical vibrations are related to the energies of the corresponding state by the usual quantum expressions E' = hv' and E" = hv" so that E' - E" = h (v'-v") which is one of the Bohr's postulates.

Critical Potential (Excitation and Ionisation Potential) In Bohr's theory, it is assumed that an atom can exist only in a definite number of states each of which is characterised by a definite amount of energy. Generally an atom exists in a state of least potential energy. In any gas discharge tube, electrons from the cathode pass through the gas ,and collide with the atoms of gas in a manner which may be of two types : (i) One type is elastic collision in which energy and momentum of colliding electron remain conserved and (ii) the other type is inelastic collision in which a part of energy of colliding electron is used up in raising its internal energy. If the energy gained by the electron is equal to or greater than the energy necessary to raise the atom from its nonnal state to the state of higher energy, the atom is said to be in the excited state. In this state, the atom remains for a very short time (10-8 sec) and possesses the tendency to return to the normal state with the emission of absorbed energy in the form of radiation. The frequency of emitted radiation is given by Bohr's theory hv = Eexclted state-Enormal state The minimum kinetic energy required to produce this effect is known as excitation energy, i.e., Exciting energy = Yzmu 2 = (e) (Ve) ergs where Ve is the excitation potential which is required to accelerate the electron from rest. As there are a large number of energy levels in an atom, there will be number of excitation potentials. If we apply the Bohr's theory, the excitation energy from 1st to nth state is given by Ali ::::; Rhc(~ - _1_) ergs 2

e

Rhc

n

Ali = 1.602 x 10-12

(

1 1)ev 12--;r (0.39)

When n = 2, 3 etc., we will get the successive excitation potentials from equation (0.39) by putting these values of n. If the energy supplied to an atom is sufficient to remove the electron from the influence of an atom the energy required is known as ionisation energy and the potential through which a colliding electron is raised is known as ionisation potential. For this case, n = 00, equation (0.39) becomes as

Concepts in Spectroscopy

1.35

M

= 13.61(_1__ ~) = 13.61 ev. (1)2

Therefore, ionisation potential

=

00

(0.40)

13.61 ev.

Vector Atom Model We know that each electron has an orbital quantum number I and spin quantum number s. Both these quantum numbers are vector quantities. The angular momenta and magnetic dipole moments which are also vector quantities are calculated by using the values of I and s. If an atom is having many electrons, the orbital quantum numbers of each electron are to be added vectorially to obtain the total orbital quantum number. Thus, the total orbital quantum number, L, may be defmed as follows :

where m is the number of electrons. Similarly, the total spin quantum number, S, may be defined as follows :

.

[1=1

Fig. 0.21 : Sum of Orbital Quantum Numbers m

S=LSi j=\

If II = 2 and 12 = I, then L = 11+1 and the values of L can be 3, 2 and 1 (shown in Fig. 1.21). The angular momenta will be .,j12hI21t, {6h121t and .,j2hl21t and the resultant angular momentum will be as follows: GL =

~[L(L+I)]h 21t

JUh J6h J2h

=--,--or-21t 21t 21t

Thus, there are three possible ways of combining the two vectors to get the three different values for the resultant. Whenever, the spin quantum numbers of different electrons are to be combined, one has to take Hund's rule into consideration. Thus S = sl+s2 will, be either 0 or 1 (Sl = s2 = 12). The spin of each electron is a half integral value but the resultant spin of the two electrons will be a full integral. The angular momentum because of the resultant spin would be Gs = .,j[S(S+ 1)] = 0 or .,j2hI21t ; the angular momentum of each electron was

J3~. This is shown in Fig. 0.22. 2 21t

1.36

Concepts in Spectroscopy

-. 8,=% Fig. 0.22 : Sum of Spin Quantum Numbers

Similarly the values of resultant magnetic dipole moments are calculated. The dipole moment because of the orbital quantum number woufd be ..,j[L(L+1)]pB

eh

= ..,j[L(L+1)].-and that because of the 47tm

spin quantum number would be

eh 47tm In general, total quantum number J is given as follows : J=L+S, the total angular momentum is given as follows : GJ = ..,j[J(J+l)]hl27t, and the total magnetic dipole moment is as follows : Total magnetic dipole moment = ..,j[J(J+1)]. g. pB. where g is known as Lande splitting factor and its value is given as follows : 2..,j[S(S+1)]. J1B

=

2..,j[S(S+1)].

J(J + 1) +S(S+ 1) -(L+ 1) g = 1+

2J(J +1)

We also have to remember the following rules; (i) The vector sum of the orbital quantum numbers of all the electrons present in a closed shell is always zero, i.e.,

'l

(ii)

L = L = 0 The vector sum of the spin quantum numbers of all the electrons present in a closed shell is zero, i.e.,

S = L Si = C From the rules (i) and (ii), it means that the total angular momentum of all the electrons present in a closed shell is zero and also the total magnetic dipole moment is zero. From these rules it also follows that the electrons forming a closed shell do not interact with the other atoms and also with the external magnetic induction. Such is the case with rare gases. Because of what is explained above, we need not consider the electrons forming the closed shell when we are calculating the total angular momentum. Only the electrons in the outermost shell need be considered. For example, when we consider the optical properties of lithium (2s electron), sodium (3s electron) and potassium (4s electron), we will consider only the electrons in the highest shell. Similarly, when we consider the optical properties of beryllium, magnesium, calcium etc., we have to consider only the two s electrons in the outermost shell. The orbital angular momentum vector l and spin angular momentum vector s combine in several ways to give the resultant vector which represent the atom in vector model. The commonly known two schemes of addition are L-S coupling and j-j coupling. These are the limiting cases of the whole range of permutation and combination of land s for the electrons of a complex atonl.

Concepts in Spectroscopy

1.37

1. L-S Coupling : In this coupling scheme, all the angular momentum vectors I of the electrons combine to form the resultant L. Similarly and independently all their spin angular momentum vectors s combine to form the resultant S. These two resultant vectors add vectorically to give the total angular momenta of the atom. This may be symbolically represented as follows:

(1\+/2+/3+",) + (s\+s2+s3+"') = L+S = J The total quantum number J can take all the positive integral values from L +S to L-S or S-L depending upon whether L>S or S>L. The L--S couping is possible when the interaction between individual orbital momenta on one hand and the individual spin angular momenta on the other hand are very strong. This coupling is also called normal coupling or Russell Saunders coupling.

2. j-j Coupling: In this coupling scheme the interaction between the orbital angular momentum and spin angular momentum in each electron is stronger than that of the orbital or the spin vectors of the other electrons in the atom. Hence, the contribution to the total angular momentum due to several electrons of the same atom is individual contribution from each electron of the atom. If the total angular momentum of a single electron in a multielectron atom is j = I + s then the total angular momentum of the atom due to several electrons may be given by =

(I\+s\) + (12 + s2) + (l3+s3) + ... + (In+sn)

=

j \ + j2 + j3 + .. ·jn J

=

This type of coupling is experienced very rarely and for majority of the cases L-S coupling is effective. Notations. For a single electron the states are represented as s, p, d, f, g, h, ... when I = 0, 1, 2, 3, 4, 5, ... respectively. When several electrons are present in the atom depending upon the total quantum number L the notations given are S, P, D, F, G, H, ....... when L = 0, 1, 2, 3, 4, 5, respectively. Further, the multiplicity of an energy level is given by (2S+ 1). This number is given on the left side above as a subscript. On the right hand side after the letter denoting the value of L, the value of J is written below the level of the value L. On the left hand side of the whole is written the value of n, the principal quantum number. Thus, the 32S\/2 means n = 3, L = 0, S = Yz, multiplicity (2S+1) = 2 and J = Yz ; 3 2P3/2 means n = 3, L = 1, S = Yz, multiplicity (2S+1) = 2 and J = 3/2. The 3 3D3 means n = 3, L = 2, S = 1, multiplicity (2S+ 1) = 3 and J = 3. Example 1. Give the special notation for the following states of the atom. (i) n

=

4, L = 2, S =

°(ii) n

=

4, L

=

1, S = 1, J

=

°and (iii) n

=

3, L = 2 and multiplicity 2.

Solution. (i) Multiplicity (2S+ 1) = 1 ; J = L+S = 2 :. State will be 4 \D 2. (ii)

Multiplicity (2S+ 1) = 3

(iii)

Multiplicity will be (2S+ 1) = or S = Yz

:. State will be 4

3p0

As L = 2, J = 5/2 or 3/2. The two states which are positive are 3 2Ds/2 or 3 2D3/2 Spectra of Alkali Metals Like the hydrogen atom, the spectra of the alkali metals consist of a series of lines with regularly

1.38

Concepts in Spectroscopy

. decreasing separation and decreasing intensity. One important feature of the alkali spectra is that the lines are mostly doublets though each and every line has not been resolved into two in the spectroscope. Quantum defect. The spectral lines obtained from alkali metals cannot be represented by a formula exactly analogous to Balmer's formula. As the lines converge to a limit, one is able to represent them by the difference of two spectral terms. nus the values of the terms are not of Balmer's form i.e., Ri n2 but of the form RJ*n 2, where n is the prinCipal ,quantum number and n* is the effective quantum number and its value is given by (n+o.), where a. is a small quantity and is known the quantum defect. The magnitude of the quantum defect depends upon the .penetration of the core by the orbits of the emitting electron. Increased penetration will lead to negative value of a.. We also know that the amount of penetration of the core depends upon the shape of the orbit. It, is, therefore, expected that a series of different values of a. will occur for different values. of azimuthal quantum number. There are four types of quantum defect (a.) designated by S, P, D, and F. These words stand for the fIrst letter of sharp, principal, diffuse and fundamental. The names of the series are attributed according to the order, as to which of them appears in the quantum defect of,the running term. The principal series is most easily excited and is hence so called. The lines in the sharp series are well defIned and those of the diffuse series are blurred. In emission spectra, other series in addition to the principal series may be observed for the alkali metals. These series partly overlap each other. The last named series viz, the fundamental of Bergmann series, occur in the infra red. However, the frequencies of the lines of the four series may consequently be represented by the following general expressions :

Diffuse series

v = aP-nS v = as-nP v aP-nD

Fundamental series

v

Sharp series Principal series

=

=

aD - nF.

where a is a constant, and n is the principal quantum number. Explanation: As earlier stated, the value of n* is given by (n+o.). When we calculate the values of nand n * for the S, P, D, and F terms of different alkali metals, we get the following values :

Lithium

Sodium

Term Series

n*( = n+a)

n

S

1.59

2

P

2

D

1.96 3.00

F

4.00

3 4 3

0 2 3 0

S

1.63

P

2.12

3

D

2.99

3

2

F

4.00

4

3

From the above table we fInd that the discrepancies between n* and n decrease in the order of S, P, D, and F of term series. The discrepancies [n*-n = (n+o.)-n = a.], also called quantum defect have been explained as follows. If we compare the structure of an alkali metal with hydrogen, it may be regarded that its nucleus is a single unit and around which the additional electron rotates. This additional electron is known as optical electron. In hydrogen, the electron remains in its fixed orbits and therefore, definite spectral lines will be observed. In the case of an alkali metal, the optical electron does not always rotate in its own orbits. It

Concepts in Spectroscopy

1.39 F 6 5

Fig. 0.23 : Energy Level Diagram of Sodium

has a tendency to penetrate into the other inner orbits and this penetration depends upon the eccentricity of the orbit. The greater the eccentricity of the orbit, the more closely the electron may approach the nucleus in the course of its rotation. It is, therefore more likely that greater the penetration of the inner levels, the greater will be divergence from hydrogen like behaviour. A simplified energy level diagram for sodium is shown in Fig. (0.23); those for the other alkali metals are similar, except for the difference in the principal quantum number of the optical electrons. The integers indicate the various values of the principal quantum number n in each spectral state, represented by S, P, D and F. It may be noted that the S, P, D and F terms are actually doublets but this is not shown in the figure. Explanation of the Doublet Structure When we study the spectra of elements other than hydrogen we observe that many lines are actually multiples consisting of two, three or more lines close together. In order to explain Uhlenbeck and Goldsmit (1925) suggested that this multiplicity of spectral lines is due to the spin of the electron. The contribution due to the spin is quantised and is expressed in terms of spin quantum number. The value of this number is +~ and -~. The value of the spin and azimuthal quantum number is given by } = I+s. where 'j' is called the inner quantum number. As's' can have +~ and -~, it follows that for every value of 'f' there are two values of 'j' viz .. } 1 = I + ~ and j2 = I Yz The above values hold good except when I is zero. In that case the two 'j' values are identical, i. e., +~ and -~, because it is not the +ve or -ve sign but is the numerical value only of 'j' which determines the momentum. It means that every 'I' level except I = 0 is consequently split into two levels with different energies. Let us now illustrate this discussion by applying to alkali metals. (i) For an electron in an'S' level, the value of 1 = o. It means that the two values of} are numerically identical and hence it is a singlet level

}I = }I =

l+~ o+~

andh = andh =

l-~ o-~

}I=+~andh=+~

Concepts in Spectroscopy

1.40 (ii)

In the P level, I

=

1 and so the corresponding values of j' are 2/3 and 1,/2.

1+1,/2 and j2 =1-1,/2 = 1+1,/2 andj2 = 1-1,/2 jl = 3/2 h = 1,/2 Thus, each spectral line in 'P' level is doublet. Similarly in the D and F levels the values of j' are 512 and 712 respectively. Again j' has two values and therefore each line will be a doublet. The doublet of the sharp series of spectra be represented by the expressions jl j2

~

=

= aP 1I2-nS and

~

= aP312-nS

where S is a singlet and P is a doublet. Photo Electric Effect Introduction: In 1887, Hertz conducted a series of experiments on the production of electromagnetic waves by oscillating charges. The oscillations were initiated by a spark jumping across a gap between metal electrodes which were connected to a circuit containing inductance and capacitance. However, in course of his experiments, Hertz made the accidental discovery that the detector gap length could be increased without preventing a :;park from occurring, provided the transmitter spark was visible from the detector gap. He discovered that it was ultraviolet light from the transmitter gap which was effective in permitting a lengthening of the detector gap.

RC

Fig. 0.24 : Experimental Arrangement for Demonstrating the Photoelectric Effect

In 1888, Hallwachs made the important observation that ultraviolet light falling on a charged plate caused it, under some circumstances, to lose its charge. The principle of the experiment performed by Hallwachs is illustrated in Fig. 0.24. He connected a negatively charged zinc plate to an electroscope and found that the plate lost its charge when ultraviolet light was made to fall upon it. It was evidently the result of the release of electrons by the zinc plate under the influence of the ultraviolet light. It was further found that whereas certain metals like zinc, magnesium, lithium etc., respond to ultraviolet light, some alkali metals like sodium, potassium and rubidium were sensitive even to visible light. These results are now known to be due to the photoelectric effect. Definition. It is the phenomenon of ejection of electrons from a metal plate when light of a suitable wa·"e-length falls on it. The electrons emitted are called photo = hv-hvo

(3)

It is clear from the above equation that the maximum velocity of photo----electrons will increase with increase in frequency v of incident radiation and no electrons will be emitted when v is less than vo' Testing : The Einstein's equation explains the observed facts about photo----electric emission. This equation was tested in 1915 by Millikan who observed its results in agreement with experimental results. The successful explanation of photo----electric effect by Einstein on the basis of quantum theory offers one of the evidences in favour of the quantum theory. Importance : Photo----electric effect is used in the construction of photoelectric cell which converts light energy into electric energy. The photo----electric cells are widely used in various fields.

Photoelectric Cell It is a device for converting light energy into electric energy. There are three main types of photocells. 1. Photo-emission Type: It consists of an evacuated glass, of quartz tube which has its inner surface M coated with sodium, potassium or caesium, and this coating is connected to-ve end of a battery. A window is left open through which light can enter the tube, and a wire loop P, or a cylindrical wire in the centre is connected to the +ve end of the battery and it collects the electrons (Fig. 0.26). The two parts are insulated from each other and are brought out to two legs at the bottom of the photo cell.

Concepts in Spectroscopy

1.42

On admitting light, which is of a frequency above the threshold frequency, electrons from the photometal are emitted; they travel towards the wire loop and a current is produced in the circuit. Since a photocell responds to the light which falls on it, as does the human eye, it is therefore called an electric eye. There are two types of photo emission cells, (1) high vacuum type, and (2) gas filled type. In vacuum cells, the photoelectric current is very small but these cells keep a strict proportionality between the current and intensity of light. The chief advantages of vacuum type are : (i) The sensitivity of these cells remain unaltered for a very long time provided the cathode is properly selected. . (ii)

There is no time lag between the incident light and the photoelectrons and the photoelectric, current is proportional to the intensity of illumination. Thus, they are extremely accurate in response.

Light

R

- IIIIII + Battery Fig. 0.26 : Photo Emission Cells

Fig. 0.27 : Photo Conductive Cells

Due to the above mentioned advantage, these cells are being used in television and photometry. In gas filled type cells, the response to light is not so quick but since an inert gas is being used in the cell, ionisation of the gas is produced by the emitted electrons which therefore increase the photo current. Photoelectric current in this case is not proportional to the intensity of illumination and there is some time lag. Such cells are used in cinematography both for recording and reproduction of sound. 2. Photo-conductive Cell : These cells are based on the property that the resistance of selenium and certain other metals decreases with increase of illumination. No photo-electrons are emitted in this case. A photo-conductive cell consists of a thin film of a semiconductor, e.g., selenium, B, placed below a thin semitransparent metal film A. The combination is placed on a block of iron C in contact with it (Fig. 0.27). Ordinary selenium is a semiconductor. When light is incident on A, it is absorbed by the selenium layer, electrons are emitted and if a small external e.mf. is applied, a current flows in the resistance and can operate a relay when the intensity of incident light is high. These cells are generally connected directly to microammeter or a low resistance relay and are preferably used in Wheatstone bridge arrangements. Their response to light is not as quick as in photo-emission cell, but they are more sensitive to red light. 3. Photo Voltaic Cell: This type of photo-cell consists of a thin layer of cuprous oxide coated on a disc of copper (Fig. 0.28). The oxide film has sputtered silver or gold film on its upper surface. When light falls on the oxide, electrons are emitted from it not into the surrounding air but into the copper. The oxide layer thus becomes +vely charged and copper negatively. An e.mf. is thus set up. If contacts are made to the sputtered film and to the copper, the photo voltaic e.m.f. produces a current. Photo-voltaic cells are used as exposure meters or light intensity meters. Fig. 0.28 : Photovoltalc Cells Applications of Photoelectric cells : Main applications are :

Concepts in Spectroscopy

1.43

(1) Reproduction of sound in films, (2) In televisions (3) In meterology as day-light recorders, (4) For measuring the complexion of persons. (5) In automatic light switching for switching on and switching off the street light. (6) In determination of temperature of stars. (7) In burglar alarms to detect thieves and in fire alarms to indicate the out breaks of fire, (8) In traffic signals. (9) As thermostat to keep the temperature of a furnace constant. Compton Effect (i)

From the classical theory of X-ray scattering, one is aware of the following conclusions : The scattered X-ray should possess the same wavelength as the incident one.

(ii)

The scattering constant (denoted by cr) should have a constant value 0,2 for all types of X-ray scatterings. It means that the scattering constant should be independent of the wavelength of the incident radiation.

(iii)

The distribution of intensity of X-rays should be symmetrical in the scattered X-rays.

When the experimental results from the research of scattering of hard X-rays and y-rays were analysed, the following points of discrepancy from classical expectation were brought out : (i)

The scattering radiation was found to have a greater wavelength than that of the incident radiation.

(ii)

The scattering coefficient cr was found to vary with the wavelength of the incident radiation and its value is diminished as the wavelength of the incident radiation is decreased.

(iii)

The distribution of scattered radiation was not symmetrical. Scattering was taking place in the same direction as that of the incident radiation.

In order to explain the above experimental facts about the scattering of X-rays, Compton in 1925 proposed that the phenomenon of scattering might be regarded as an elastic collision of two particles, photon of X-rays and the scatterer. When a photon of energy hv collides with an electron, it will lose a part of its energy in the form of kinetic energy to the electron. Thus, the scattered photon will have a smaller energy hv' and, in consequence, a lower frequency or greater wavelength than that of the incident photon. This observed change in radiation frequency or the wavelength of the scattered is known as Compton effect. In such a case an electron is !llso ejected from the scatterer with an energy depending upon its direction. This ejected electron is known as Compton recoil electron

Incident Photon Electron

!& e : o~

::~~ :

i:

..

"'Q'

~o~

0"

Fig. 0.29 : Compton Effect

Theory : Compton calculated mathematically the increase in wavelength by making use of quantum theory. Suppose the incident X-rays consist of photons each of energy hv, where v is the frequency. This energy (hv) is much greater than that required to eject a free eleotron from the target or scattere!. Let the photon collide with an electron at rest in the target. A part of its energy is imparted to the electron which is ejected with a velocity u in a direction making an angle 9 with that of the incident X-ray photon (Fig. 0.29). The remaining energy is associated with the X-ray photon, having lower frequency v' and moving in direction making an angle cp with that of the incident photon. If mo is the rest mass of the electron, its mass, when moving with a velocity u, will be given by the theory of relativity as :

1.44

Concepts in Spectroscopy

m=H

(i)

where e is the veiocity of light, On applying the principle of conservation of energy, we get hv + m oc 2 = hv' + me2 or m'c2 = h( v-v ') + moc2 Squaring this above equation we get m 2c4 = h2(v-2vv'+v'2)+2h(v-v')moc 2+mo2c4

(2)

On applying the principle of conservation of momentum in the direction of incident photon, we get hv -

c

hv c

=-cos~+mu

cosS

(3)

and in a direction perpendicular to the direction of the incident photon

hv , ~ , S O=~sm'l'-mu sm

c Equations (3) and (4) on simplification yield muc cos e = h(v-v' cos cf» and muc sin = hv' sin cf>, Squaring and adding equations (5) and (6), we get m 2u 2c 2 (cos 2 9 + sin2 9) = h 2 (v-v' cos cf»2 + h 2v'2 sin2 or m 2u2c 2 = h 2 (v2-v,2 cos2 cf> - 2vv' cos cf> + v'2 sin2 cf» = h 2 (v2-2vv' cos cf> + v'2)

(4)

(5)

e

(6) cf>

(7)

Subtracting equation (7) from (2), we get m 2c 2 (c 2_u 2) = -2vv' h2 (i- cos cf» + 2h (v-v,) m oc 2 + m o2c4

or

or

2h (v-v') m2c2 = 2vv'h 2 (i-cos cf»

v-v

[using eq. (i)]

h(i-cos~)

or

i i h - - - =--(i-cos~) v v m oc 2

or

---=-(i-cos~)

c

c

h

v

v

moc

(8)

If A and A' are the wavelengths corresponding to the frequencies v and v' respectively, then equation (8) may be put as

Concepts in Spectroscopy

1.45

h A'-A = DoA = -(l-cos~)

(9)

moc

From the equation (9), it follows that the increase in wavelength DoA is independent of the wavelength of the incident radiations and the nature of the scattering substance, but depends upon the angle of scattering. The quantity hlmoc is known as Compton wavelength. But

cosljJ

=2

sin2

1.2 = 1

Therefore, equation (9) becomes

A'-A = DoA =

or

A' = A +

211

.

--Sill

moc

2h

.

--Sill

moc

2

2 ~ -

2

cb -

(10)

2

The following three cases may arise (i)

When the scattering angle is zero, IjJ is zero and then, sin

1.2 = sin 0 = 0

Therefore, equation (10) becomes as

A' = A

(11 )

Thus, there is no scattering. (ii)

When 8 = 90°, sin 2 8/2 = 112 ; equation (10) becomes as

A'- A =

h (12)

-

moc

As h. mo and c are constants, equation (12) becomes as

Slits ......................... ........................

II II

Ka only

Unscattered X-Rays

·· . .. ,,/

"

...,.,,-//' Fig. 0.30 : Schematic Diagram for the Experimental Demonstration of Compton Effect

1.46

Concepts in Spectroscopy 2h

'A,'- k

= constant = - moc

2h

or (iii)

When

/),,'A,

e=

= constant = - -

(13)

moc

180°, sin2 180/2 = 1; equation (10) becomes as 2h

'A,'- 'A, = -

(14)

moc

Experimental Demonstration of Compton Effect. The schematic representation of the experimental set up used to study the Compton effect is given in Fig. 0.30. A beam of monochromatic X-rays is obtained from a source and after confining it through narrow, slits, is made to fall upon a scattering substance. The radiations scattered at some