{"id":322,"date":"2018-11-16T09:34:46","date_gmt":"2018-11-16T09:34:46","guid":{"rendered":"http:\/\/phyp10.epgpbooks.inflibnet.ac.in\/?post_type=chapter&p=322"},"modified":"2019-04-30T08:34:23","modified_gmt":"2019-04-30T08:34:23","slug":"non-linear-raman-effect","status":"publish","type":"chapter","link":"https:\/\/ebooks.inflibnet.ac.in\/phyp10\/chapter\/non-linear-raman-effect\/","title":{"rendered":"Non-linear Raman Effect"},"content":{"raw":"
\"epgp<\/a>\r\n<\/span><\/div>\r\n
\r\n\r\n \r\n\r\nContents:<\/strong>\r\n\r\n \r\n\r\n1.\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 <\/strong>Stimulated Raman Effect<\/strong>\r\n\r\n2.\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 <\/strong>Hyper Raman effect<\/strong>\r\n\r\n3.\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 <\/strong>CARS<\/strong>\r\n\r\n4.\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 <\/strong>Surface Enhanced Raman Scattering\/Spectroscopy (SERS)<\/strong>\r\n\r\n5.\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 <\/strong>Luminescence<\/strong>\r\n\r\n6.\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 <\/strong>Raman Effect Versus Fluorescence<\/strong>\r\n\r\n7.\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 <\/strong>Spectra of Polyatomic Molecules<\/strong>\r\n\r\n \r\n

The students will able to learn about <\/strong>Stimulated Raman Effect , Hyper Raman effect , CARS, Surface Enhanced Raman Scattering\/Spectroscopy (SERS) , Luminescence, Raman Effect Versus Fluorescence, Spectra of Polyatomic Molecules<\/p>\r\n \r\n

1. Stimulated Raman effect<\/strong><\/p>\r\n \r\n

A number of molecules from the initial state Ej are excited to the final state Ef when the incident light intensity becomes very large, and the Raman lines are also intense in comparison to that of ordinary light source. The molecules thus undergo two simultaneous interactions; one by the original light source with the frequency \u03c9L and then by the light originated from the Stokes line with frequency \u03c9s = \u03c9L \u2013 \u03c9v or the Anti-stokes lines \u03c9as = \u03c9L + \u03c9v where \u03c9v is the vibrational frequency of the molecule.<\/span><\/p>\r\n \r\n

Thus if a Raman active material is subjected to an interaction with a high power laser like switched Ruby laser, the high power pulse induces gain in the medium at Stoke\/Anti-Stokes frequencies \u03c9L \u00b1 n \u03c9v (n = 1, 2, 3\u2026.) shifted from\u00a0<\/span>the laser frequency \u03c9L. A strong coherent light is build up at the shifted frequency, provided the gain is high enough to compensate for cavity loss. This parametric interaction leads to an energy exchange between the pump wave and the Stokes or Anti-stokes that further lead to parametric amplification. The power transferred to the first stokes line \u03c9s = \u03c9L \u2013 \u03c9v related exponentially to the laser power at \u03c9L. Hence the first stokes line rapidly become intense enough to act as a powerful source and give rise to another stokes line<\/span><\/p>\r\n \r\n

(\u03c9L \u2013 \u03c9v) \u2013 \u03c9v = \u03c9L - 2\u03c9v<\/span><\/p>\r\n\r\n<\/div>\r\n

\r\n\r\n \r\n

This line is further intense enough to act as a powerful source and give rise to a third stokes line and so on... However, the generation of an anti-stokes line does not arise as a downward transition from a populated upper state.<\/p>\r\n \r\n

The differences between the normal Raman Effect (linear)and the stimulate Raman effect (nonlinear) are the following:<\/p>\r\n\r\n<\/div>\r\n\"\"\r\n

\r\n\r\n2. Hyper Raman effect<\/strong>\r\n\r\n \r\n

When a system is illuminated by an intense (from a giant pulse laser) radiation \u03c9L scattered radiations are obtained at frequencies 2 \u03c9L and at 2 \u03c9L = \u03c9v, (inelastic scattering). In this case, the intensity of the scattered radiations depends on the square of the laser intensity. These elastic and nonelastic scattering are now known as hyper-Rayleigh and Hyper Raman scatterings<\/p>\r\n \r\n

3. Coherent Anti- Stokes Raman Scattering (CARS)<\/strong><\/p>\r\n \r\n

It has now become possible to increase the efficiency of Raman scattering by using near- resonant incident radiation using tunable lasers. . The fundamental frequencies can be doubled and sum- and difference frequencies are obtained using frequency mixing originating from the nonlinear terms in the polarizability. There is a mixing of three frequencies originating from the third order polarizability in CARS wherein two lasers of frequencies \u03c91 and \u03c92 are chosen in such a way that their difference \u03c91 \u2013 \u03c92 concides with a Raman active vibrational frequency of the molecule \u03c9v.<\/span><\/p>\r\n \r\n

These incident waves are considered as the pump wave and the Stokes wave. The advantage is that the Stokes wave at \u03c92 is already present as a part of stimulated Raman scattering to produce a large population density in the vibrational excited state. This medium acts as a nonlinear medium for the generation of antistokes radiation at \u03c9a = 2\u03c91 \u2013 \u03c92 by the incident wave with a frequency \u03c91. Similarly, Stokes lines with frequency \u03c9s = 2\u03c92 \u2013 \u03c91 is generated by the incident waves \u03c91 and \u03c92. It is to note that four waves are involved in CARS.<\/span><\/p>\r\n \r\n

The molecules of anisotropic liquid, when placed in an electric field tend to align themselves parallel to the direction of the field. As the molecules are not symmetrical, this causes the liquid to be anisotropic and birefringent. Birefringence results in splitting of light waves into two components with different velocities that causes different refractive indices for differently polarized light. This phenomenon of electrically induced birefringence in anisotropic liquids is called Kerr effect. The birefringence and hence the change in refractive index is proportional to the external field. This property is used in electro \u2013 optic shutters. When the birefringence is linear to electric field (applied to certain crystals) the phenomenon is called Pockel\u2019s effect.<\/span><\/p>\r\n\r\n<\/div>\r\n

\r\n\r\n \r\n\r\n4.\u00a0\u00a0\u00a0\u00a0 <\/strong>Surface Enhanced Raman Scattering\/Spectroscopy (SERS)<\/strong>\r\n\r\n \r\n

As the name suggest this is a Surface-sensitive technique that enhances Raman scattering by molecules adsorbed on rough metal surfaces. The enhancement factor is as high as 1014<\/sup> \u2013 1015<\/sup>, that allows the technique to be sensitive enough to detect single molecules.<\/p>\r\n \r\n

There are two primary theories, though their mechanisms differ substantially.<\/p>\r\n \r\n

The electromagnetic theory proposes the excitation of localized surface plasmons and on the other hand, the formation of charge transfer complexes by the chemical theory is proposed.<\/p>\r\n \r\n

The chemical theory applies only for species that forms a chemical bond with the surface, so it cannot explain the observed signal enhancement in all cases, whereas the electromagnetic theory can apply even in those cases where the specimen is adsorbed only to the surface. It has been shown recently that SERS enhancement can occur even when an excited molecule is relatively far apart from the surface which hosts metallic nanoparticles enabling surface plasmon phenomena.<\/p>\r\n \r\n

5.\u00a0\u00a0\u00a0\u00a0 <\/strong>Luminescence<\/strong><\/p>\r\n \r\n

When the molecules of a gas are illuminated with light of a definite frequency, these go to an excited electronic state and these revert to their initial state with the emission of discrete radiation of frequencies which is smaller than the frequency of the absorbed light. This phenomenon is known\u00a0as\u2019 luminescence\u2019. In case the emission vanishes immediately after the removal of the exciting radiation, the phenomenon is termed as \u2019fluorescence\u2019, otherwise it is termed as \u2019phosphorescence\u2019 in case persists for an appreciable time. Their combined nomenclature is luminescence.<\/span><\/p>\r\n \r\n

Fluorescence<\/strong><\/p>\r\n \r\n

In an absorption experiment, at room temperature, the <\/span>v<\/em>\u2019\u2019 = 0 level of the ground electronic state of is most populated and thus illuminating the molecules with continuous light, a single progression of bands in the electronic absorption spectrum due to transitions form v\u2019\u2019= 0 , so v\u2019 varying from 0 to large values.<\/span><\/p>\r\n\r\n<\/div>\r\n

\r\n\r\n\"\"\r\n\r\n<\/div>\r\n
\r\n\r\n \r\n

For I2 like molecules which are heavy molecules, absorption transitions corresponding to v\u2019\u2019=1,2, may also be observed, though weak. The intensity distribution in these bands depends upon the shapes and relative positions of the two potential energy curves that is governed by Franck-Condon principle. The absorption spectra display the pattern of vibrational levels upto the limit of dissociation and the dissociation energy of the molecule in the excited electronic state can be deduced.<\/p>\r\n \r\n

There are now at least two different processes by which the excited molecule can lose its excess energy and return to its ground state. When the excess energy may be lost as heat by repeated collisions with neighbouring molecules, no emission of radiation is observed in this process that is referred to \u2019non-radiative \u2019 process. In case, the excited molecule return directly to the level v\u2019\u2019=0 of the ground state with emission of absorbed light, or , to other levels of the ground state with emission of light of lower frequencies, it is termed as fluorescence .<\/p>\r\n \r\n

Consider that a diatomic gas at low pressure ( at low pressure , there are meager chances of molecular collisions) is illuminated with light having the frequency of a single absorption band, the absorbing molecules are excited to the vibrational level of the upper electronic state of this particular absorption band only. The excited molecules then go over to the different vibrational levels of the ground electronic state with emission of radiation.\u00a0Thus, in the fluorescence spectrum there is a single progression (\u03bd\u1fbd=constant) of bands extending from the position of the absorption band toward lower frequencies with decreasing band separation.\u00a0<\/span><\/p>\r\n \r\n

The molecules have not lost any energy in collisions before re-emission and thus termed as \u1fferesonance fluorescence\u1fbd. The whole series of band is called the \u1fferesonance series\u1fbd and the band coinciding with the exciting absorption band is called the \u1fferesonance band \u1fbd and the such series of bands is observed in I2 like molecules.<\/span><\/p>\r\n\r\n<\/div>\r\n

\r\n\r\n\"\"\r\n\r\n \r\n\r\n \r\n

There may be fluorescence band that has v\u2019\u2019 smaller than that of the resonance ( exciting ) band. These bands lie on the high \u2013 frequency side of the resonance band and are referred as anti \u2013 Stokes members of the resonance series.<\/p>\r\n\r\n<\/div>\r\n\"\"\r\n

\r\n\r\n \r\n

Collisions become frequent if the gas is at a high pressure and in this case, the molecules excited to a particular vibrational level of the upper electronic state collide with the neighbouring molecules of the fluorescing gas and lose or gain vibrational energy thus switching over to other vibrational levels of the upper state .Transitions now takes place from all these levels to the various levels of the ground electronic state and the fluorescence spectrum consist of more v\u2019\u2019- progressions corresponding to various \u03bd\u1fbd values. It is noted that the absorption spectrum shows only one progression. Hence fluorescence is a more powerful technique than absorption.<\/p>\r\n \r\n

Now considring the fluorescence in solutions, the bulk of the molecules are in the lowest vibrational level v\u2019\u2019=0 of the ground electronic state. The\u00a0molecules are excited to various vibrational levels of the excited electronic state on absorption of radiation. In the absorption spectrum, the lowest \u2013 frequency band is the (0,0) band , and the spectrum extends to higher frequencies.<\/span><\/p>\r\n\r\n<\/div>\r\n

\r\n\r\n\"\"\r\n\r\n \r\n

The excited molecules lose vibrational energy due to collisions and most of them eventually come down to the lowest level \u03bd\u1fbd=0 of the excited electronic state before the fluorescence occurs. This is a non-radiative process and is called \u1ffevibrational deactivation\u1fbd. Fluorescent emission then take place from v\u2019\u2019=0 level to the various levels of the ground electronic state and occur within ~ 10-4 second after excitation .The fluorescence spectrum witnesses the (0,0) band is the band of highest frequency and the spectrum extends to lower frequencies. The absorption spectrum displays the vibrational level of the excited electronic state while; the fluorescence spectrum displays those of the ground state.<\/p>\r\n\r\n<\/div>\r\n

\r\n\r\n \r\n\r\nPhosphorescence<\/strong>\r\n\r\n \r\n

Phosphorescence is a delayed emission and persists for periods up to seconds after the absorption process is ended. It is understood as a result from transitions that connect electronic states of different multiplicities.<\/p>\r\n \r\n

The figure depicts the mechanism of phosphorescent emission. Initially, the molecules of a sample are mostly in the lowest vibrational level v\u2019\u2019=0 of the ground electronic state in case of diatomic molecules ( except O2) is a singlet state. These are excited to another singlet electronic state upon absorption of radiation. Phosphorescence arises when a triplet excited state of the molecule exists between the singlet excited state and the ground state , and its potential energy curve crosses the curve of the excited singlet state as shown in the figure. The excited molecules can undergo radiationless transitions to a lower vibrational level due to collisions at isoenergic states of singlets and triplets. This is again a non-radiative process known as \u1ffeinternal conversion\u1fbd and the molecule is at triplet state. Further, vibrational deactivation by collisions in the triplet state takes the molecule stepwise down the vibrational levels until it reaches the lowest level of the triplet state.Transitions from this triplet state to the ground singlet state are responsible for the Phosphorescence. These transitions are however, forbidden due to the spin selection rule (\u2206S=0) . Thus these transitions have very long half lives, and the resulting Phosphorescent radiation is emitted seconds or even minutes after the initial absorption .<\/p>\r\n \r\n

There is a possibility of the intermediate (crossing) electronic state if it is also a singlet state then the above process occurs very quickly and leads to fluorescent emission.<\/span><\/p>\r\n\r\n<\/div>\r\n

\r\n\r\n\"\"\r\n\r\n<\/div>\r\n
\r\n

Hence, both fluorescence and phosphorescence are the emissions due to the de-excitation of the molecules from the excited electronic states to the ground state; fluorescence arises from transitions between electronic states of the same multiplicity while the Phosphorescence arises from transitions between states of different multiplicities.<\/p>\r\n\r\n<\/div>\r\n

\r\n\r\n6. Raman Effect Versus Fluorescence:<\/strong>\r\n\r\n \r\n

Raman Effect and fluorescence resemble each other because in each case light suffers change of wavelength after falling on a scatterer. However, Raman Effect differs from fluorescence as:<\/p>\r\n \r\n

1.In fluorescent spectrum, the frequencies are always less than the incident frequency In Raman spectrum lines with higher frequencies i.e anti-Stokes lines and lower frequencies (Stokes lines) are observed.<\/p>\r\n

2. In case of Fluorescence, frequencies of scattered radiation is independent of frequencies of exciting radiation<\/p>\r\n \r\n

In Raman spectroscopy the scattered radiation is directly related to the incident radiation frequency.<\/p>\r\n \r\n

3.Fluorescent lines are unpolarized<\/p>\r\n

Raman lines are strongly polarized<\/p>\r\n \r\n

4. Fluorescence is due to real electronic transitions<\/p>\r\n

Raman scattering is due to virtual electronic transitions<\/p>\r\n5. Raman spectrum is more informative than fluorescence spectrum\r\n\r\n6.\u00a0 Fluorescence is an absorption and re-emission process\r\n\r\nRaman scattering is an inelastic scattering process\r\n\r\n \r\n\r\n7.\u00a0 Spectra of Polyatomic Molecules<\/strong>\r\n\r\n \r\n

The values of force constants are to be calculated from the observed vibrational frequencies. Though there is only one vibrational frequency for diatomic molecules, but the vibrations depend upon the number of atoms in a polyatomic\u00a0molecule. If a molecule contains <\/span>n<\/em> atoms, there is a need to specify by <\/span>3n<\/em> coordinates i.e. three coordinates for each atom. The potential energy being a function of the relative position of the atoms, one can conveniently transform into internal coordinates and separate out the motion of center of mass. The center of mass has three degrees of freedom (translational motion) and thus leaves <\/span>3n<\/em> - 3 internal degrees of freedom for the molecule. There are either three or two rotational degrees of freedom depending on whether the molecule is linear or nonlinear.<\/span><\/p>\r\n \r\n

For a linear molecule, <\/span>3n<\/em> - 5 vibrations exist while for non-linear molecule 3n-6 vibrations exist, <\/span>n<\/em> being the number of atoms in the molecule.<\/span><\/p>\r\n \r\n

Considering a polyatomic molecule, water, H20, the number of vibrational frequencies expected (3 x 3)-6 = 3. Experimentally due to the complexity of the spectrumor otherwise due to symmetry consideration, these frequencies may not be observable sometimes. But, by a combined effort with infrared and Raman spectra, the fundamental frequencies can be identified. A theoretical calculation of the frequencies and the amplitudes of motion can be made by formulating the kinetic energy and potential energy of the molecule in matrix form. These calculated values of the amplitudes of vibrations can be compared with the experimental values obtained by other methods like electron diffraction.<\/span><\/p>\r\n\r\n<\/div>\r\n\"\"\r\n

\r\n\r\n \r\n

The fundamental frequencies of vibration obtained from infrared and Raman spectra provide considerable information about the interatomic forces in various molecules.<\/p>\r\n \r\n

It is to note that different types of valence bonds exhibit different degrees of resistance to stretching and bending that are roughly independent of the molecule in which the bond occurs.<\/p>\r\n \r\n

Further there are empirical relations between the length of a bond and its resistance to stretching that gives useful results.<\/p>\r\n \r\n

Nuclear magnetic resonance spectroscopy (NMR) also yields a lot of information in this direction where microwave and infrared are extensively used in such studies. The preferred relative orientation of portions of molecules connected with single bonds has proved to be of decisive consequence in the activity of important biological molecules and has posed a challenge to theoretical chemists to attempt an explanation of the system.<\/p>\r\n \r\n

Although infrared, Raman and magnetic resonance studies are quite useful but the experimental tools like microwave and gas phase electron diffraction techniques\u00a0are most helpful. The study of the molecular energy levels can provide a number of physical and chemical properties of molecules and the information thus obtained is very important for a chemist, a biologist and an astrophysicist or a space physicist and even in Forensic laboratories.<\/span><\/p>\r\n \r\n

Vibration<\/strong><\/p>\r\n \r\n

It have seen that the number of internal vibrations of a molecule with <\/span>n<\/em> atoms is<\/span><\/p>\r\n\r\n<\/div>\r\n

\r\n\r\n \r\n\r\n3n <\/em>- 5 for linear molecules and\r\n\r\n3n <\/em>- 6 for non-linear molecules\r\n\r\n \r\n

In the case of linear or non- linear molecules with n<\/em> atoms there are n<\/em> - I bonds between its atoms and hence n<\/em> - I of the above vibrations are bond stretching and the remaining 2n<\/em> - 5 (non-linear) and 2n<\/em> - 4 (linear) are bending motions.<\/p>\r\n \r\n

In the case of a diatomic molecules that is a particular case of linear molecule, the number of constituting atoms are 2 and hence the number of vibrations 3n<\/em> - 5 = I and hence there can be only one fundamental vibration. The presence of vertones is -due to the anharmonocity of the molecule. These vibrations are normal modes of vibrations. A normal coordinate, is a single coordinate along which the normal mode of a vibration can be followed. The vibration does not cause a translation of the center of mass of the molecule. It is a linear combination of the motions of the individual atoms. If we have a simple diatomic molecules 'H79<\/sup>Br, the 'H nucleus will move 79\/1 times fast as that of 79<\/sup>Br.<\/p>\r\n \r\n\r\nq=r-r0= \u2206r(H)+\u2206r(Br)\r\n\r\nand mH<\/sub>\u2206r(H)=mBr<\/sub>\u2206r(Br)\r\n\r\n\u2206r(Br)\/ \u2206r(H)= mH<\/sub>\/ mBr<\/sub>\r\n\r\n<\/div>\r\n

Mathematically the normal coordinates are expressed such that the potential energy <\/span>V= 1\/2 \u03bb<\/em>i<\/em>q<\/em>2<\/em>i<\/em> where<\/span> \u03bb <\/em>is a constant and Kinetic Energy<\/span> K <\/em>= (\u00bd) mi<\/span>(dq<\/em>i<\/em>\/dt)2.<\/em><\/p>\r\n \r\n

If we consider a nonlinear triatomic molecule of the type <\/span>AX<\/em>2<\/em><\/sub>,<\/em> there are <\/span>3n<\/em> - 6 = 3 vibrations.<\/span><\/p>\r\n \r\n

All three atoms vibrate with the same frequency in a normal mode. The example for an <\/span>AX<\/em>2<\/em> type is the water molecule with the chemical formula H2<\/sub>0.<\/span><\/p>\r\n \r\n

Among the three possibly vibrations two are symmetric and one is anti symmetric. From the figure of symmetric vibrations it can be seen that if the molecule is rotated in an axis which bisects the HOH angle the appearance of the vibrations look alike before and after this kind of rotation. Such kind of vibrations are called symmetric vibrations and the axis of rotation is called a C2 axis as by rotating about 360\/2 = 180 degrees the molecule still have the same appearance and by rotating twice the molecule attains the original position. It is clear from the figure that the <\/span>v<\/em>1<\/em> (symmetric stretch) and <\/span>v<\/em>2<\/em> (symmetric bending) are symmetric vibrations and <\/span>v<\/em>3<\/em> is an anti- symmetric stretch.<\/span><\/p>\r\n \r\n

Further all these vibrations are infrared active. For an infrared active, there must be a dipole change during the vibration. The dipole change may be parallel or perpendicular to the symmetry axis. In the case of symmetric vibrations in H20 , the dipole changes along the direction of the symmetry axis C2 where as in the case of anti symmetric stretching the dipole change is perpendicular to this axis.\u00a0<\/span><\/p>\r\n \r\n

CO2 molecule is a linear triatomic molecule and hence the number of vibrations are <\/span>3n-<\/em>5=4. There are two sets of symmetry axes. In this molecule, the carbon atom occupies the middle position and has a two-fold axis C2 perpendicular to the bond\u00a0<\/span>axis and an infinite fold axis C passing through the bond axis itself. It is called C as by rotation of the molecule about this axis in any angle gives identical appearance of the molecule. The different kinds of vibrations in the CO2 molecules are shown in the figure<\/span><\/p>\r\n\"\"\r\n

<\/div>\r\n
\r\n\r\n \r\n

In the case of v<\/em>1<\/em> symmetric stretch there is no dipole change due to vibration and hence it is not infrared active. Among the four vibrational modes expected, the v<\/em>2<\/em> bending mode consists of two vibrations one in the plane of the paper and the other perpendicular. The Oxygen atoms of CO2 molecule, move perpendicular to the plane i.e. into and out of the plane. These are called degenerate vibrations as these are two independent vibrations and the frequency of these vibration are the same.<\/p>\r\n \r\n

When the molecules are considered as anharmonic oscillators rather than harmonic, several overtones of the type 2v<\/em>1<\/em>,<\/em> 3v<\/em>1<\/em>, 4v<\/em>1<\/em>,<\/em> ... or 2v<\/em>2<\/em>,<\/em> 3v<\/em>2<\/em> ... , 2v<\/em>3<\/em>,<\/em> 3v<\/em>3<\/em> ... etc or combination bands like v<\/em>1<\/em>+ v<\/em>2<\/em>,<\/em> 2v<\/em>1 + v<\/em>2<\/em> ... etc or difference bands of the type v<\/em>1 - v<\/em>2<\/em>, v<\/em>1 + v<\/em>2<\/em> - v<\/em>3<\/em> ... etc also observed.<\/p>\r\n\r\n<\/div>\r\n

\r\n\r\nRotation of a Polyatomic Molecule<\/strong>\r\n\r\n \r\n

The rotational selection rule for a polyatomic molecule depend on the nature of vibration, i.e either it is parallel or perpendicular \u2013 or in other worlds, the dipole change along or parallel to the symmetry axis or perpendicular to the symmetry axis. The selection rules for a parallel vibration in the case of a linear Molecule (assuming that the molecule undergoes anharmonic oscillation) are<\/p>\r\n \r\n\r\n\u0394\u03bd<\/em>=\u00b1I,\u00b12,\u00b13...and\r\n\r\n \r\n\r\n\u0394J= <\/em>\u00b1 I\r\n\r\n \r\n

Thus R and P<\/em> branches are obtained as in case of certain diatomic molecules. The lines are equally spaced ignoring centrifugal distortion and without having line occurring at the band center. As B<\/em> value is comparatively smaller in the case of polyatomic molecules, the spacing of the lines in the two branches is quite small. In heavier molecules, the rotational constants are smaller and therefore Rand P<\/em> contours are observed. An estimate of the rotational constant can be derived from the separation of the two contours.<\/p>\r\n \r\n\r\nFor a perpendicular vibration of a linear molecule the selection rules are\r\n\r\n\u0394\u03bd<\/em>=\u00b1I,\u00b12,\u00b13...and\r\n\r\n\u0394J=0, <\/em>\u00b1 I\r\n\r\n \r\n

Therefore, there are three branches of lines R, P<\/em> and a Q<\/em> branch. Q<\/em> branches are superimposed at the origin and are intense.<\/p>\r\n \r\n

The molecules with zero dipole moment do not show any pure rotational spectrum but show infrared spectrum. If the spectra exhibits rotationally resolved structure one could obtain the rotational constants. In the case of linear polyatomic molecules like CO2<\/sub> or C2<\/sub>H2<\/sub> with a center of symmetry, the rotational spectrum observed in the Raman Spectrum is very much influenced by the nuclear pins. In the case of CO2<\/sub> alternate rotational lines are missing as the nuclear spin of the oxygen is zero and in the case of acetylene <\/span>the<\/em> intensity ratio of nearby lines are 3:I, thus the <\/span>P<\/em> and <\/span>R<\/em> branch lines show strong, weak, strong, weak ... lines.<\/span><\/p>\r\n \r\n

In the case of a symmetric top molecule, the energy for a rotation vibration line may be represented by (neglecting centrifugal distortion)<\/span><\/p>\r\n \r\n

E<\/em>\u03bd,J<\/em> = Evibrational + Erotational =<\/span> \u03c9<\/em>e<\/em>x<\/em>e<\/em>(v<\/em>+<\/span>1\/2) <\/em>-<\/span> \u03c9<\/em>e<\/em>(<\/span>v<\/em>+<\/span>1\/2)<\/em>2<\/em> + ...<\/span> BJ(J<\/em>+I) +<\/span> (A<\/em>-<\/span>B)K<\/em><\/p>\r\n\r\n<\/div>\r\n

\r\n\r\n \r\n

Here again the change of dipole moment parallel to the symmetry axis gives rise to parallel bands and the dipole change perpendicular to the symmetry axis gives rise to perpendiecular bands. The selection rules for the parallel vibrations are<\/p>\r\n

\u0394v <\/em>= \u00b1 I, \u00b12, \u00b13 ete and<\/p>\r\n \r\n\r\nJ <\/em>= 0, \u00b1 I, \u0394K <\/em>= 0\r\n\r\n \r\n\r\nAnd for perpendicular bands\r\n\r\n \r\n\r\nv <\/em>= \u00b1 I , \u00b1 2, \u00b1 3 etc and\u00a0\u00a0 J <\/em>= 0, \u00b11, \u0394K <\/em>= \u00b1 1\r\n\r\n \r\n\r\nThus in both cases we have R, P<\/em> and Q<\/em> branches of lines.\r\n\r\n<\/div>\r\n \r\n\r\nSummary<\/strong>\r\n\r\n \r\n\r\nStimulated Raman Effect<\/strong>\r\n\r\nHyper Raman effect<\/strong>\r\n\r\nCARS<\/strong>\r\n\r\nSurface Enhanced Raman Scattering\/Spectroscopy (SERS)<\/strong>\r\n\r\nLuminescence<\/strong>\r\n\r\nRaman Effect Versus Fluorescence<\/strong>\r\n\r\nSpectra of Polyatomic Molec<\/strong>\r\n\r\n\r\n\r\n\r\n
you can view video on Non-linear Raman Effect<\/strong><\/td>\r\n\"\"<\/a><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>","rendered":"
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Contents:<\/strong><\/p>\n

 <\/p>\n

1.\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 <\/strong>Stimulated Raman Effect<\/strong><\/p>\n

2.\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 <\/strong>Hyper Raman effect<\/strong><\/p>\n

3.\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 <\/strong>CARS<\/strong><\/p>\n

4.\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 <\/strong>Surface Enhanced Raman Scattering\/Spectroscopy (SERS)<\/strong><\/p>\n

5.\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 <\/strong>Luminescence<\/strong><\/p>\n

6.\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 <\/strong>Raman Effect Versus Fluorescence<\/strong><\/p>\n

7.\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 <\/strong>Spectra of Polyatomic Molecules<\/strong><\/p>\n

 <\/p>\n

The students will able to learn about <\/strong>Stimulated Raman Effect , Hyper Raman effect , CARS, Surface Enhanced Raman Scattering\/Spectroscopy (SERS) , Luminescence, Raman Effect Versus Fluorescence, Spectra of Polyatomic Molecules<\/p>\n

 <\/p>\n

1. Stimulated Raman effect<\/strong><\/p>\n

 <\/p>\n

A number of molecules from the initial state Ej are excited to the final state Ef when the incident light intensity becomes very large, and the Raman lines are also intense in comparison to that of ordinary light source. The molecules thus undergo two simultaneous interactions; one by the original light source with the frequency \u03c9L and then by the light originated from the Stokes line with frequency \u03c9s = \u03c9L \u2013 \u03c9v or the Anti-stokes lines \u03c9as = \u03c9L + \u03c9v where \u03c9v is the vibrational frequency of the molecule.<\/span><\/p>\n

 <\/p>\n

Thus if a Raman active material is subjected to an interaction with a high power laser like switched Ruby laser, the high power pulse induces gain in the medium at Stoke\/Anti-Stokes frequencies \u03c9L \u00b1 n \u03c9v (n = 1, 2, 3\u2026.) shifted from\u00a0<\/span>the laser frequency \u03c9L. A strong coherent light is build up at the shifted frequency, provided the gain is high enough to compensate for cavity loss. This parametric interaction leads to an energy exchange between the pump wave and the Stokes or Anti-stokes that further lead to parametric amplification. The power transferred to the first stokes line \u03c9s = \u03c9L \u2013 \u03c9v related exponentially to the laser power at \u03c9L. Hence the first stokes line rapidly become intense enough to act as a powerful source and give rise to another stokes line<\/span><\/p>\n

 <\/p>\n

(\u03c9L \u2013 \u03c9v) \u2013 \u03c9v = \u03c9L – 2\u03c9v<\/span><\/p>\n<\/div>\n

\n

 <\/p>\n

This line is further intense enough to act as a powerful source and give rise to a third stokes line and so on… However, the generation of an anti-stokes line does not arise as a downward transition from a populated upper state.<\/p>\n

 <\/p>\n

The differences between the normal Raman Effect (linear)and the stimulate Raman effect (nonlinear) are the following:<\/p>\n<\/div>\n

\"\"<\/p>\n

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2. Hyper Raman effect<\/strong><\/p>\n

 <\/p>\n

When a system is illuminated by an intense (from a giant pulse laser) radiation \u03c9L scattered radiations are obtained at frequencies 2 \u03c9L and at 2 \u03c9L = \u03c9v, (inelastic scattering). In this case, the intensity of the scattered radiations depends on the square of the laser intensity. These elastic and nonelastic scattering are now known as hyper-Rayleigh and Hyper Raman scatterings<\/p>\n

 <\/p>\n

3. Coherent Anti- Stokes Raman Scattering (CARS)<\/strong><\/p>\n

 <\/p>\n

It has now become possible to increase the efficiency of Raman scattering by using near- resonant incident radiation using tunable lasers. . The fundamental frequencies can be doubled and sum- and difference frequencies are obtained using frequency mixing originating from the nonlinear terms in the polarizability. There is a mixing of three frequencies originating from the third order polarizability in CARS wherein two lasers of frequencies \u03c91 and \u03c92 are chosen in such a way that their difference \u03c91 \u2013 \u03c92 concides with a Raman active vibrational frequency of the molecule \u03c9v.<\/span><\/p>\n

 <\/p>\n

These incident waves are considered as the pump wave and the Stokes wave. The advantage is that the Stokes wave at \u03c92 is already present as a part of stimulated Raman scattering to produce a large population density in the vibrational excited state. This medium acts as a nonlinear medium for the generation of antistokes radiation at \u03c9a = 2\u03c91 \u2013 \u03c92 by the incident wave with a frequency \u03c91. Similarly, Stokes lines with frequency \u03c9s = 2\u03c92 \u2013 \u03c91 is generated by the incident waves \u03c91 and \u03c92. It is to note that four waves are involved in CARS.<\/span><\/p>\n

 <\/p>\n

The molecules of anisotropic liquid, when placed in an electric field tend to align themselves parallel to the direction of the field. As the molecules are not symmetrical, this causes the liquid to be anisotropic and birefringent. Birefringence results in splitting of light waves into two components with different velocities that causes different refractive indices for differently polarized light. This phenomenon of electrically induced birefringence in anisotropic liquids is called Kerr effect. The birefringence and hence the change in refractive index is proportional to the external field. This property is used in electro \u2013 optic shutters. When the birefringence is linear to electric field (applied to certain crystals) the phenomenon is called Pockel\u2019s effect.<\/span><\/p>\n<\/div>\n

\n

 <\/p>\n

4.\u00a0\u00a0\u00a0\u00a0 <\/strong>Surface Enhanced Raman Scattering\/Spectroscopy (SERS)<\/strong><\/p>\n

 <\/p>\n

As the name suggest this is a Surface-sensitive technique that enhances Raman scattering by molecules adsorbed on rough metal surfaces. The enhancement factor is as high as 1014<\/sup> \u2013 1015<\/sup>, that allows the technique to be sensitive enough to detect single molecules.<\/p>\n

 <\/p>\n

There are two primary theories, though their mechanisms differ substantially.<\/p>\n

 <\/p>\n

The electromagnetic theory proposes the excitation of localized surface plasmons and on the other hand, the formation of charge transfer complexes by the chemical theory is proposed.<\/p>\n

 <\/p>\n

The chemical theory applies only for species that forms a chemical bond with the surface, so it cannot explain the observed signal enhancement in all cases, whereas the electromagnetic theory can apply even in those cases where the specimen is adsorbed only to the surface. It has been shown recently that SERS enhancement can occur even when an excited molecule is relatively far apart from the surface which hosts metallic nanoparticles enabling surface plasmon phenomena.<\/p>\n

 <\/p>\n

5.\u00a0\u00a0\u00a0\u00a0 <\/strong>Luminescence<\/strong><\/p>\n

 <\/p>\n

When the molecules of a gas are illuminated with light of a definite frequency, these go to an excited electronic state and these revert to their initial state with the emission of discrete radiation of frequencies which is smaller than the frequency of the absorbed light. This phenomenon is known\u00a0as\u2019 luminescence\u2019. In case the emission vanishes immediately after the removal of the exciting radiation, the phenomenon is termed as \u2019fluorescence\u2019, otherwise it is termed as \u2019phosphorescence\u2019 in case persists for an appreciable time. Their combined nomenclature is luminescence.<\/span><\/p>\n

 <\/p>\n

Fluorescence<\/strong><\/p>\n

 <\/p>\n

In an absorption experiment, at room temperature, the <\/span>v<\/em>\u2019\u2019 = 0 level of the ground electronic state of is most populated and thus illuminating the molecules with continuous light, a single progression of bands in the electronic absorption spectrum due to transitions form v\u2019\u2019= 0 , so v\u2019 varying from 0 to large values.<\/span><\/p>\n<\/div>\n

\n

\"\"<\/p>\n<\/div>\n

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 <\/p>\n

For I2 like molecules which are heavy molecules, absorption transitions corresponding to v\u2019\u2019=1,2, may also be observed, though weak. The intensity distribution in these bands depends upon the shapes and relative positions of the two potential energy curves that is governed by Franck-Condon principle. The absorption spectra display the pattern of vibrational levels upto the limit of dissociation and the dissociation energy of the molecule in the excited electronic state can be deduced.<\/p>\n

 <\/p>\n

There are now at least two different processes by which the excited molecule can lose its excess energy and return to its ground state. When the excess energy may be lost as heat by repeated collisions with neighbouring molecules, no emission of radiation is observed in this process that is referred to \u2019non-radiative \u2019 process. In case, the excited molecule return directly to the level v\u2019\u2019=0 of the ground state with emission of absorbed light, or , to other levels of the ground state with emission of light of lower frequencies, it is termed as fluorescence .<\/p>\n

 <\/p>\n

Consider that a diatomic gas at low pressure ( at low pressure , there are meager chances of molecular collisions) is illuminated with light having the frequency of a single absorption band, the absorbing molecules are excited to the vibrational level of the upper electronic state of this particular absorption band only. The excited molecules then go over to the different vibrational levels of the ground electronic state with emission of radiation.\u00a0Thus, in the fluorescence spectrum there is a single progression (\u03bd\u1fbd=constant) of bands extending from the position of the absorption band toward lower frequencies with decreasing band separation.\u00a0<\/span><\/p>\n

 <\/p>\n

The molecules have not lost any energy in collisions before re-emission and thus termed as \u1fferesonance fluorescence\u1fbd. The whole series of band is called the \u1fferesonance series\u1fbd and the band coinciding with the exciting absorption band is called the \u1fferesonance band \u1fbd and the such series of bands is observed in I2 like molecules.<\/span><\/p>\n<\/div>\n

\n

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 <\/p>\n

 <\/p>\n

There may be fluorescence band that has v\u2019\u2019 smaller than that of the resonance ( exciting ) band. These bands lie on the high \u2013 frequency side of the resonance band and are referred as anti \u2013 Stokes members of the resonance series.<\/p>\n<\/div>\n

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 <\/p>\n

Collisions become frequent if the gas is at a high pressure and in this case, the molecules excited to a particular vibrational level of the upper electronic state collide with the neighbouring molecules of the fluorescing gas and lose or gain vibrational energy thus switching over to other vibrational levels of the upper state .Transitions now takes place from all these levels to the various levels of the ground electronic state and the fluorescence spectrum consist of more v\u2019\u2019- progressions corresponding to various \u03bd\u1fbd values. It is noted that the absorption spectrum shows only one progression. Hence fluorescence is a more powerful technique than absorption.<\/p>\n

 <\/p>\n

Now considring the fluorescence in solutions, the bulk of the molecules are in the lowest vibrational level v\u2019\u2019=0 of the ground electronic state. The\u00a0molecules are excited to various vibrational levels of the excited electronic state on absorption of radiation. In the absorption spectrum, the lowest \u2013 frequency band is the (0,0) band , and the spectrum extends to higher frequencies.<\/span><\/p>\n<\/div>\n

\n

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 <\/p>\n

The excited molecules lose vibrational energy due to collisions and most of them eventually come down to the lowest level \u03bd\u1fbd=0 of the excited electronic state before the fluorescence occurs. This is a non-radiative process and is called \u1ffevibrational deactivation\u1fbd. Fluorescent emission then take place from v\u2019\u2019=0 level to the various levels of the ground electronic state and occur within ~ 10-4 second after excitation .The fluorescence spectrum witnesses the (0,0) band is the band of highest frequency and the spectrum extends to lower frequencies. The absorption spectrum displays the vibrational level of the excited electronic state while; the fluorescence spectrum displays those of the ground state.<\/p>\n<\/div>\n

\n

 <\/p>\n

Phosphorescence<\/strong><\/p>\n

 <\/p>\n

Phosphorescence is a delayed emission and persists for periods up to seconds after the absorption process is ended. It is understood as a result from transitions that connect electronic states of different multiplicities.<\/p>\n

 <\/p>\n

The figure depicts the mechanism of phosphorescent emission. Initially, the molecules of a sample are mostly in the lowest vibrational level v\u2019\u2019=0 of the ground electronic state in case of diatomic molecules ( except O2) is a singlet state. These are excited to another singlet electronic state upon absorption of radiation. Phosphorescence arises when a triplet excited state of the molecule exists between the singlet excited state and the ground state , and its potential energy curve crosses the curve of the excited singlet state as shown in the figure. The excited molecules can undergo radiationless transitions to a lower vibrational level due to collisions at isoenergic states of singlets and triplets. This is again a non-radiative process known as \u1ffeinternal conversion\u1fbd and the molecule is at triplet state. Further, vibrational deactivation by collisions in the triplet state takes the molecule stepwise down the vibrational levels until it reaches the lowest level of the triplet state.Transitions from this triplet state to the ground singlet state are responsible for the Phosphorescence. These transitions are however, forbidden due to the spin selection rule (\u2206S=0) . Thus these transitions have very long half lives, and the resulting Phosphorescent radiation is emitted seconds or even minutes after the initial absorption .<\/p>\n

 <\/p>\n

There is a possibility of the intermediate (crossing) electronic state if it is also a singlet state then the above process occurs very quickly and leads to fluorescent emission.<\/span><\/p>\n<\/div>\n

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\"\"<\/p>\n<\/div>\n

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Hence, both fluorescence and phosphorescence are the emissions due to the de-excitation of the molecules from the excited electronic states to the ground state; fluorescence arises from transitions between electronic states of the same multiplicity while the Phosphorescence arises from transitions between states of different multiplicities.<\/p>\n<\/div>\n

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6. Raman Effect Versus Fluorescence:<\/strong><\/p>\n

 <\/p>\n

Raman Effect and fluorescence resemble each other because in each case light suffers change of wavelength after falling on a scatterer. However, Raman Effect differs from fluorescence as:<\/p>\n

 <\/p>\n

1.In fluorescent spectrum, the frequencies are always less than the incident frequency In Raman spectrum lines with higher frequencies i.e anti-Stokes lines and lower frequencies (Stokes lines) are observed.<\/p>\n

2. In case of Fluorescence, frequencies of scattered radiation is independent of frequencies of exciting radiation<\/p>\n

 <\/p>\n

In Raman spectroscopy the scattered radiation is directly related to the incident radiation frequency.<\/p>\n

 <\/p>\n

3.Fluorescent lines are unpolarized<\/p>\n

Raman lines are strongly polarized<\/p>\n

 <\/p>\n

4. Fluorescence is due to real electronic transitions<\/p>\n

Raman scattering is due to virtual electronic transitions<\/p>\n

5. Raman spectrum is more informative than fluorescence spectrum<\/p>\n

6.\u00a0 Fluorescence is an absorption and re-emission process<\/p>\n

Raman scattering is an inelastic scattering process<\/p>\n

 <\/p>\n

7.\u00a0 Spectra of Polyatomic Molecules<\/strong><\/p>\n

 <\/p>\n

The values of force constants are to be calculated from the observed vibrational frequencies. Though there is only one vibrational frequency for diatomic molecules, but the vibrations depend upon the number of atoms in a polyatomic\u00a0molecule. If a molecule contains <\/span>n<\/em> atoms, there is a need to specify by <\/span>3n<\/em> coordinates i.e. three coordinates for each atom. The potential energy being a function of the relative position of the atoms, one can conveniently transform into internal coordinates and separate out the motion of center of mass. The center of mass has three degrees of freedom (translational motion) and thus leaves <\/span>3n<\/em> – 3 internal degrees of freedom for the molecule. There are either three or two rotational degrees of freedom depending on whether the molecule is linear or nonlinear.<\/span><\/p>\n

 <\/p>\n

For a linear molecule, <\/span>3n<\/em> – 5 vibrations exist while for non-linear molecule 3n-6 vibrations exist, <\/span>n<\/em> being the number of atoms in the molecule.<\/span><\/p>\n

 <\/p>\n

Considering a polyatomic molecule, water, H20, the number of vibrational frequencies expected (3 x 3)-6 = 3. Experimentally due to the complexity of the spectrumor otherwise due to symmetry consideration, these frequencies may not be observable sometimes. But, by a combined effort with infrared and Raman spectra, the fundamental frequencies can be identified. A theoretical calculation of the frequencies and the amplitudes of motion can be made by formulating the kinetic energy and potential energy of the molecule in matrix form. These calculated values of the amplitudes of vibrations can be compared with the experimental values obtained by other methods like electron diffraction.<\/span><\/p>\n<\/div>\n

\"\"<\/p>\n

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 <\/p>\n

The fundamental frequencies of vibration obtained from infrared and Raman spectra provide considerable information about the interatomic forces in various molecules.<\/p>\n

 <\/p>\n

It is to note that different types of valence bonds exhibit different degrees of resistance to stretching and bending that are roughly independent of the molecule in which the bond occurs.<\/p>\n

 <\/p>\n

Further there are empirical relations between the length of a bond and its resistance to stretching that gives useful results.<\/p>\n

 <\/p>\n

Nuclear magnetic resonance spectroscopy (NMR) also yields a lot of information in this direction where microwave and infrared are extensively used in such studies. The preferred relative orientation of portions of molecules connected with single bonds has proved to be of decisive consequence in the activity of important biological molecules and has posed a challenge to theoretical chemists to attempt an explanation of the system.<\/p>\n

 <\/p>\n

Although infrared, Raman and magnetic resonance studies are quite useful but the experimental tools like microwave and gas phase electron diffraction techniques\u00a0are most helpful. The study of the molecular energy levels can provide a number of physical and chemical properties of molecules and the information thus obtained is very important for a chemist, a biologist and an astrophysicist or a space physicist and even in Forensic laboratories.<\/span><\/p>\n

 <\/p>\n

Vibration<\/strong><\/p>\n

 <\/p>\n

It have seen that the number of internal vibrations of a molecule with <\/span>n<\/em> atoms is<\/span><\/p>\n<\/div>\n

\n

 <\/p>\n

3n <\/em>– 5 for linear molecules and<\/p>\n

3n <\/em>– 6 for non-linear molecules<\/p>\n

 <\/p>\n

In the case of linear or non- linear molecules with n<\/em> atoms there are n<\/em> – I bonds between its atoms and hence n<\/em> – I of the above vibrations are bond stretching and the remaining 2n<\/em> – 5 (non-linear) and 2n<\/em> – 4 (linear) are bending motions.<\/p>\n

 <\/p>\n

In the case of a diatomic molecules that is a particular case of linear molecule, the number of constituting atoms are 2 and hence the number of vibrations 3n<\/em> – 5 = I and hence there can be only one fundamental vibration. The presence of vertones is -due to the anharmonocity of the molecule. These vibrations are normal modes of vibrations. A normal coordinate, is a single coordinate along which the normal mode of a vibration can be followed. The vibration does not cause a translation of the center of mass of the molecule. It is a linear combination of the motions of the individual atoms. If we have a simple diatomic molecules ‘H79<\/sup>Br, the ‘H nucleus will move 79\/1 times fast as that of 79<\/sup>Br.<\/p>\n

 <\/p>\n

q=r-r0= \u2206r(H)+\u2206r(Br)<\/p>\n

and mH<\/sub>\u2206r(H)=mBr<\/sub>\u2206r(Br)<\/p>\n

\u2206r(Br)\/ \u2206r(H)= mH<\/sub>\/ mBr<\/sub><\/p>\n<\/div>\n

Mathematically the normal coordinates are expressed such that the potential energy <\/span>V= 1\/2 \u03bb<\/em>i<\/em>q<\/em>2<\/em>i<\/em> where<\/span> \u03bb <\/em>is a constant and Kinetic Energy<\/span> K <\/em>= (\u00bd) mi<\/span>(dq<\/em>i<\/em>\/dt)2.<\/em><\/p>\n

 <\/p>\n

If we consider a nonlinear triatomic molecule of the type <\/span>AX<\/em>2<\/em><\/sub>,<\/em> there are <\/span>3n<\/em> – 6 = 3 vibrations.<\/span><\/p>\n

 <\/p>\n

All three atoms vibrate with the same frequency in a normal mode. The example for an <\/span>AX<\/em>2<\/em> type is the water molecule with the chemical formula H2<\/sub>0.<\/span><\/p>\n

 <\/p>\n

Among the three possibly vibrations two are symmetric and one is anti symmetric. From the figure of symmetric vibrations it can be seen that if the molecule is rotated in an axis which bisects the HOH angle the appearance of the vibrations look alike before and after this kind of rotation. Such kind of vibrations are called symmetric vibrations and the axis of rotation is called a C2 axis as by rotating about 360\/2 = 180 degrees the molecule still have the same appearance and by rotating twice the molecule attains the original position. It is clear from the figure that the <\/span>v<\/em>1<\/em> (symmetric stretch) and <\/span>v<\/em>2<\/em> (symmetric bending) are symmetric vibrations and <\/span>v<\/em>3<\/em> is an anti- symmetric stretch.<\/span><\/p>\n

 <\/p>\n

Further all these vibrations are infrared active. For an infrared active, there must be a dipole change during the vibration. The dipole change may be parallel or perpendicular to the symmetry axis. In the case of symmetric vibrations in H20 , the dipole changes along the direction of the symmetry axis C2 where as in the case of anti symmetric stretching the dipole change is perpendicular to this axis.\u00a0<\/span><\/p>\n

 <\/p>\n

CO2 molecule is a linear triatomic molecule and hence the number of vibrations are <\/span>3n-<\/em>5=4. There are two sets of symmetry axes. In this molecule, the carbon atom occupies the middle position and has a two-fold axis C2 perpendicular to the bond\u00a0<\/span>axis and an infinite fold axis C passing through the bond axis itself. It is called C as by rotation of the molecule about this axis in any angle gives identical appearance of the molecule. The different kinds of vibrations in the CO2 molecules are shown in the figure<\/span><\/p>\n

\"\"<\/p>\n

<\/div>\n
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 <\/p>\n

In the case of v<\/em>1<\/em> symmetric stretch there is no dipole change due to vibration and hence it is not infrared active. Among the four vibrational modes expected, the v<\/em>2<\/em> bending mode consists of two vibrations one in the plane of the paper and the other perpendicular. The Oxygen atoms of CO2 molecule, move perpendicular to the plane i.e. into and out of the plane. These are called degenerate vibrations as these are two independent vibrations and the frequency of these vibration are the same.<\/p>\n

 <\/p>\n

When the molecules are considered as anharmonic oscillators rather than harmonic, several overtones of the type 2v<\/em>1<\/em>,<\/em> 3v<\/em>1<\/em>, 4v<\/em>1<\/em>,<\/em> … or 2v<\/em>2<\/em>,<\/em> 3v<\/em>2<\/em> … , 2v<\/em>3<\/em>,<\/em> 3v<\/em>3<\/em> … etc or combination bands like v<\/em>1<\/em>+ v<\/em>2<\/em>,<\/em> 2v<\/em>1 + v<\/em>2<\/em> … etc or difference bands of the type v<\/em>1 – v<\/em>2<\/em>, v<\/em>1 + v<\/em>2<\/em> – v<\/em>3<\/em> … etc also observed.<\/p>\n<\/div>\n

\n

Rotation of a Polyatomic Molecule<\/strong><\/p>\n

 <\/p>\n

The rotational selection rule for a polyatomic molecule depend on the nature of vibration, i.e either it is parallel or perpendicular \u2013 or in other worlds, the dipole change along or parallel to the symmetry axis or perpendicular to the symmetry axis. The selection rules for a parallel vibration in the case of a linear Molecule (assuming that the molecule undergoes anharmonic oscillation) are<\/p>\n

 <\/p>\n

\u0394\u03bd<\/em>=\u00b1I,\u00b12,\u00b13…and<\/p>\n

 <\/p>\n

\u0394J= <\/em>\u00b1 I<\/p>\n

 <\/p>\n

Thus R and P<\/em> branches are obtained as in case of certain diatomic molecules. The lines are equally spaced ignoring centrifugal distortion and without having line occurring at the band center. As B<\/em> value is comparatively smaller in the case of polyatomic molecules, the spacing of the lines in the two branches is quite small. In heavier molecules, the rotational constants are smaller and therefore Rand P<\/em> contours are observed. An estimate of the rotational constant can be derived from the separation of the two contours.<\/p>\n

 <\/p>\n

For a perpendicular vibration of a linear molecule the selection rules are<\/p>\n

\u0394\u03bd<\/em>=\u00b1I,\u00b12,\u00b13…and<\/p>\n

\u0394J=0, <\/em>\u00b1 I<\/p>\n

 <\/p>\n

Therefore, there are three branches of lines R, P<\/em> and a Q<\/em> branch. Q<\/em> branches are superimposed at the origin and are intense.<\/p>\n

 <\/p>\n

The molecules with zero dipole moment do not show any pure rotational spectrum but show infrared spectrum. If the spectra exhibits rotationally resolved structure one could obtain the rotational constants. In the case of linear polyatomic molecules like CO2<\/sub> or C2<\/sub>H2<\/sub> with a center of symmetry, the rotational spectrum observed in the Raman Spectrum is very much influenced by the nuclear pins. In the case of CO2<\/sub> alternate rotational lines are missing as the nuclear spin of the oxygen is zero and in the case of acetylene <\/span>the<\/em> intensity ratio of nearby lines are 3:I, thus the <\/span>P<\/em> and <\/span>R<\/em> branch lines show strong, weak, strong, weak … lines.<\/span><\/p>\n

 <\/p>\n

In the case of a symmetric top molecule, the energy for a rotation vibration line may be represented by (neglecting centrifugal distortion)<\/span><\/p>\n

 <\/p>\n

E<\/em>\u03bd,J<\/em> = Evibrational + Erotational =<\/span> \u03c9<\/em>e<\/em>x<\/em>e<\/em>(v<\/em>+<\/span>1\/2) <\/em>–<\/span> \u03c9<\/em>e<\/em>(<\/span>v<\/em>+<\/span>1\/2)<\/em>2<\/em> + …<\/span> BJ(J<\/em>+I) +<\/span> (A<\/em>–<\/span>B)K<\/em><\/p>\n<\/div>\n

\n

 <\/p>\n

Here again the change of dipole moment parallel to the symmetry axis gives rise to parallel bands and the dipole change perpendicular to the symmetry axis gives rise to perpendiecular bands. The selection rules for the parallel vibrations are<\/p>\n

\u0394v <\/em>= \u00b1 I, \u00b12, \u00b13 ete and<\/p>\n

 <\/p>\n

J <\/em>= 0, \u00b1 I, \u0394K <\/em>= 0<\/p>\n

 <\/p>\n

And for perpendicular bands<\/p>\n

 <\/p>\n

v <\/em>= \u00b1 I , \u00b1 2, \u00b1 3 etc and\u00a0\u00a0 J <\/em>= 0, \u00b11, \u0394K <\/em>= \u00b1 1<\/p>\n

 <\/p>\n

Thus in both cases we have R, P<\/em> and Q<\/em> branches of lines.<\/p>\n<\/div>\n

 <\/p>\n

Summary<\/strong><\/p>\n

 <\/p>\n

Stimulated Raman Effect<\/strong><\/p>\n

Hyper Raman effect<\/strong><\/p>\n

CARS<\/strong><\/p>\n

Surface Enhanced Raman Scattering\/Spectroscopy (SERS)<\/strong><\/p>\n

Luminescence<\/strong><\/p>\n

Raman Effect Versus Fluorescence<\/strong><\/p>\n

Spectra of Polyatomic Molec<\/strong><\/p>\n\n\n\n
you can view video on Non-linear Raman Effect<\/strong><\/td>\n\"\"<\/a><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n","protected":false},"author":3,"menu_order":14,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-322","chapter","type-chapter","status-publish","hentry"],"part":3,"_links":{"self":[{"href":"https:\/\/ebooks.inflibnet.ac.in\/phyp10\/wp-json\/pressbooks\/v2\/chapters\/322","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/ebooks.inflibnet.ac.in\/phyp10\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/ebooks.inflibnet.ac.in\/phyp10\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/ebooks.inflibnet.ac.in\/phyp10\/wp-json\/wp\/v2\/users\/3"}],"version-history":[{"count":6,"href":"https:\/\/ebooks.inflibnet.ac.in\/phyp10\/wp-json\/pressbooks\/v2\/chapters\/322\/revisions"}],"predecessor-version":[{"id":693,"href":"https:\/\/ebooks.inflibnet.ac.in\/phyp10\/wp-json\/pressbooks\/v2\/chapters\/322\/revisions\/693"}],"part":[{"href":"https:\/\/ebooks.inflibnet.ac.in\/phyp10\/wp-json\/pressbooks\/v2\/parts\/3"}],"metadata":[{"href":"https:\/\/ebooks.inflibnet.ac.in\/phyp10\/wp-json\/pressbooks\/v2\/chapters\/322\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/ebooks.inflibnet.ac.in\/phyp10\/wp-json\/wp\/v2\/media?parent=322"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/ebooks.inflibnet.ac.in\/phyp10\/wp-json\/pressbooks\/v2\/chapter-type?post=322"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/ebooks.inflibnet.ac.in\/phyp10\/wp-json\/wp\/v2\/contributor?post=322"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/ebooks.inflibnet.ac.in\/phyp10\/wp-json\/wp\/v2\/license?post=322"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}