Raman Spectroscopy R. Stuart Tobias
University of Minnesota Minneapolis 55455
in h0rganiC Chemistry
Of
the two spectroscopic methods commonly used for the study of molecular vibrations, Raman and infrared, the Raman effect provided most of the early experimental data as has been noted by Ferraro in a review in THIS JOURNAL on inorganic infrared spectroscopy (1). With the advent of commercial infrared spectrometers during the 19401s,the study of Raman scattering by molecular spectroscopists decreased markedly. Within the last few years, primarily as a result of improved apparatus, Raman spectroscopy has again become an important structural tool. While it is usually possible to obtain a good infrared spectrum of an organic compound and thus gain information about the molecular vibrations and stereochemistry, this is often not the case with inorganic systems. For example, it may be of interest to study the structure of a complex ion in aqueous solution, to obtain information on the strength of the metal-ligand bonds, and to examine the interaction of the ion with solvent water molecules. I t is impossible to obtain good infrared spectra of aqueous solutions over a wide frequency range because of the extensive absorption of infrared radiation by the water molecules. I n contrast, good Raman spectra can usually be obtained with little interference from the solvent waterprovidingmuchinformation on the dynamics and stereochemistry of polyatomic ions. Vibrations with frequencies as low as 100 em-' are easily studied with both liquid and powdered solid samples using the Raman technique, while quite sophisticated equipment is required to obtain infrared absorption spectra at such low frequencies. Although this very low frequency region is of little interest to organic chemists, it is here that many of the metalligand vibrations of interest to the inorganic chemist occur. Because of the very great usefulness of the technique, many of the recent applications of Raman spectroscopy have been made by inorganic and physical chemists as well as by molecular spectroscopists. The Scattering o f Light
In order to discuss problems where Raman spectra can contribute to a solution, it is helpful to examine briefly the origin of these spectra. Raman spectra can be observed when visible light is scattered by molecules in solids, liquids, or gases. The energy of the light used should not correspond to an allowed elecEDITOR'SNOTE The second part of this paper will appear in the February issue. I t will contribute further to the theme by discussing applications of Raman speetmseopy particularly useful in the field of inorganic chemistry.
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tronic or vibrational transition, that is, it should not be resonantly absorbed by the molecules. Since it is the scattered light which is of interest and this will be very low in intensity compared to the initial intensity of the light source, it is convenient to view the scattered light at 90' to the initial beam. The common form of the apparatus used for exciting Raman spectra is illustrated in Figure 1. A coiled mercury arc lamp surrounding the sample is usually used to excite the spectrum. Normally, the blue mercury line, 4358 A (22938 cm.-I), is isolated with filters interposed between the lamp and the sample. The sample should not absorb the exciting light, consequently in the past almost all Raman spectra have been obtained for colorless compounds. If the particulate model of light is used, that is, if the light beam is considered to he made up of a stream of photons of energy hue, the scattering of light may be pictured in terms of collisions of these photons with the molecules. We can imagine two types of collisions, elastic and inelastic. I n the elastic collisions, the photons neither gain energy from nor lose energy to the molecule. The scattered photons still have energy hue. This is the sucalled Rayleigh scattering. Such elastic scattering by gas molecules in the earth's atmosphere is responsible for the brightness and blueness of the sky.' It is also possible that a photon may gain energy from or lose energy to the molecule (an inelastic scattering process). This consideration, together with the requirement that energy be conserved during the collision, led Smekal to predict inelastic photon scattering five years before experimental observations of this effect were reported ( 2 ) . Thus, conservation of energy requires hvo
+ '/mooz + Ea = hu' + '/md2+ E'
(1)
The zero subscripts refer to the properties before collision, while primed quantities refer to the properties after the collision of the photon with the molecule. The photon energy is hv, the molecular translational kinetic energy is 'lzmu2, and the molecular internal
The material for this article iis taken in part from lectures at the Summer Institnte for College Teachers on Advanced Inorganic Chemistry at the University of Minnesota, June, 1965. This conference wa? made possible by s, grant from the National Science Foundation which has also sopported some of the author's research in thk area, grants NSF GP-653 and GP-5022. An interesting and elementary discussion may be found in MINNAERT,M., "Light and Calaur" Dover Publications, New York, 1954, chapter XI.
the number of molecules, n,, in the first excited vihrational level with v = 1 and vibrational energy 3/2hv1 relative to the number in the ground vibrational state, no, with vibrational energy 1/2hn is described by the Boltzmann distributionn, = noexp(- (3/2hvl - 1/2hvl)/ kT). Since hvl > kT,most of the molecules will have the minimum energy 1/2hvl. Thus we should see far more photons with energy h(vo - ul) than with h(vo v,). This can be generalized to polyatomic molecules with n normal modes of vibration where up to n pairs of new frequencies will be observed in the scattered light. Because of the nature of some vibrations, it may not be possible for the molecule to gain or lose energy associated with a particular vibrational mode. Thus the fundamental vibrations often lead to fewer than n pairs of Raman "lines" (this designation is a hold-over from the time when Raman spectra were recorded photographically) in the spectra of polyatomic molecules. It also should be noted that irradiation with photons having energy h v ~would lead to a resonant absorption of energy as shown in Figure 2. Such photons generally correspond to infrared radiation, and
+
Figure 1. Lamp and sample tubes for the excitation of Raman spectra; (a) Toronto mercury arc lomp and cell for liquids; (b) conical cell for crystal powders.
energy (electronic, vibrational, and rotational) is E. I n addition, linear momentum must be conserved in the collision. For the molecule, the momentum is mv, and the photon momentum is hv/c where c is the vela* ity of light. Let us, for simplicity, consider a head-on collision between the photon and the molecule. This reverses the direction of the photon, and conservation of momentnm requires
+ muo = -hv'/c + mu' ~ ( m= )mu' - muo = h(v' + v ) / c hv/e
If the scattered light does not undergo a large change in frequency, A (mv) = 2hv/c, approximately. For a typical experiment using blue light from a mercury arc, u = 6.9 X 1014sec-I, A(mv)
=
2 X 6.624 X
lo-"
erg sec X 6.9 X 1014see-' 3.0 X lOL0 em sec-I
hvo
+ Eo = h.' + E' = (u'
-
"0)
(2)
I n most studies of Raman scattering, the changes in the internal energy (Eo - E') result from changes in the vibrational energy of the molecules, although i t is also possible to study changes in the rotational and electronic energy of the molecules too. We can visualize the changes in vibrational energy upon the collision between a molecule and a photon by reference to Figure 2. The vibrational frequency of a diatomic molecule is designated ul, and the molecule has vibrational energy (v l/,)hur where v is the vibrational quantnm number.2 It was precisely these considerations which formed the basis of the original quantum mechanical treatment of the inelastic scattering put forth by Kramers and Heisenberg (S), again before the effect had been observed experimentally. According to this picture, we should observe photons which have decreased in frequency by ul as well as some which have increased in frequency by n . If we analyze the scattered light with a spectrometer, we should observe the frequencies vo and uo .t v,. Furthermore,
+
c
v=l
E = 312 hV,
v =O
E = I12hut
~i~~~~ 2.
At room temperature, the momentum of a hydrogen molecule is about 6.3 X 10-19 g cm sec-', so even with a light molecule like hydrogen the molecular momentnm and, therefore, the velocity will change by only a very small fraction. It follows that to a good approximation 1/2mvo2= 1 / 2 r n ~ and '~ (Eo - E')/h
IU
W
Origin of the vibrational Ramon effect.
this is the manner in which an infrared absorption spectrum is obtained. The experimental observation of the inelastic scattering of light was first reported in the context outlined above by C. V. Raman in 1928 for liquids and solutions (4, 5). I n these early experiments, the scattering was detected visually, and a simple pocket spectroscopewas used. Very shortly thereafter, Landsberg and Mandelstam reported similar inelastic scattering from studies on quartz crystals (6). For his discovery, Raman received the Nobel prize in physics in 1930. As with many methods for the determination of molecular structure, the earlier applications of Raman spectroscopy were made mostly by physicists; however the method now has become 'nother one of the very useful tools employed by the chemist in his investigations of molecular structure. The shift in emphasis from Ramau to infrared spectra in the work of molecular spectroscopists is revealed by quotations from textbooks dealing with the determination of molecular structure. I n 1949, Rice and Teller wrote "Experimentally, the investigation of light scattering is simpler than work in the infrared and thus the Re man effect has yielded more material about molecular 3 For an introductory discussion of molecular vibrational energies, see Baanoa, G . M., "The Structure of Molecules," W. A. Benjamin, Inc., New York, 1963, chapter 111.
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vibrations than any other method" (7). On the other hand, Evans, in discussing Raman spectroscopy in 1963, has remarked that for problems where either Raman or infrared spectroscopy could give satisfactory answers, "the infrared method is preferred" (8). I n a great part, the revival of interest t in Raman spectra among chemists has resulted from the development of commercial photoelectric instruments. These are ahnost E as convenient to use as recording visibleU.V. spectrophotometers, and they permit p accurate determination of the intensity as weU as the frequency of the scattered light. Figure 3 shows a photoelectrically recorded Raman spectnun for carbon tetrachloride. This 5 atom, tetrahedral molecule has (3 X 5 - 6) = 9 degrees of vibrational freedom, that is, 9 normal vibrational modes. Be-8w -400 o Raman F ~ W U ~ P C-cm-' Y cause of the rather symmetrical structure CCI, excited ~ i t hthe blue line of of a tetrahedral molecule, several ditrerent figure 3. Roman spectrum of the tetrahedral mo~ocu~e normal modes have the same frequency. he mercury arc, 4 3 5 8 i122938 cm-I). From simply knowing the molecule to be Figure 4 illustrates the interaction of a light wave tetrahedral, it can be predicted that three normal with a simple diatomic molecule where the wavelength modes will have one frequency, three more a second of the light is much larger than the molecular difrequency, two more a third frequency, and the last mensions. This p d n g light wave subjects the molenormal mode wiU have a fourth frequency.Vhus we cule to an electrical field oscillating at the frequency of might expect to see up to four ditrerent pairs of Raman the light, vo. The scattering is due almost entirely to lines vo v. centered about the exciting frequency. the displacements of the electrons under the influence of Indeed all four pairs of lines are visible in the specthe oscillating field which induces a classical, oscillating trum in Figure 3, although the highest frequency line dipole in the molecule. This dipole then radiates is split into two overlapping lines. As expected, the light in all directions except along the line of action of intensities of the lines shifted to higher frequencies are the dipole, although Figure 4 only shows the light much lower than the intensities of the lines shifted to scattered in one direction at an angle of 90" to the frequencies lower than the exciting line. The Rayincident wave. leigh scattering of the CCL molecules causes the inThe magnitude of the induced dipole moment M tense line at the exciting frequency, 22938 cm-', aldepends upon the amplitude of the light wave and the though there are also contributions to this caused by polarizability of the molecule which is a measure of scattering due to traces of dust, etc., in the sample. how readily the electrons are displaced in the field E of All other tetrahedral molecules and ions, for example, the light wave. C104-, SO4-, Re04-, CdBr4=, TiClr, SnH4, etc., exhibit similar spectra consisting of four pairs of lines. M = uE (3)
$
4
*
Principles of Raman Spectroscopy
A variety of reviews on the fundamentals of Raman spectroscopy have been written. The majority of these were produced during the earlier heyday of Raman spectroscopy and, insofar as applications and techniques are concerned, are now badly out of date (%It). Briefer reviews have appeared more recently (13-15) together with one monograph which gives a very detailed treatment of theory and modern experimental practice (16). I n many respects, the treatment of the scattering of light by a molecule is simplified by using the wave picture for the light. This "classical" theory of scattering was, interestingly, developed by Cabannes (17) after the quantum mechanical treatment of Kramers and Heisenberg. a A good introduction to the use of group theory in predicting the number of different vibrational frequencies to be expected in Raman and in infrared spectra can be found in C O T ~ NF., A,, "Chemical Applications of Group Theory," Interscience Publishers (Division of John Wiley & Sons, Inc.), 1963, chapter 9.
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Since the field fluctuates at frequency vo, we must express it in terms of an amplitude Eo and a time dependent term. For a vibrating molecule, the molecular polarizability, a, also will vary with time. With a diatomic molecule, the polarizahiity is approximately a linear function of the bond length for small amplitudes of vibration. I n terms of the displacements from the equilibrium bond length, that is, in terms of the vibrational coordinate QI, Q, is a displacement coordinate, that is, its value is zero in the equilibrium configuration. The polarizability in the equilibrium configuration of the molecule is aro and (ba/bQl)a refers to the rate of change of the polarizability with change in the bond length, evaluated at the equilibrium configurat,ion. Finally, since the molecule is vibrating with the f r e
quency ul, the displacement Q, is also a function of time. Q,
=
QP cos(2rvd)
(6)
Here QP is the maximum value of the displacement from the equilibrium bond length, that is, the vibrational amplitude.
Figure 4.
Interaction of a light wove with a diatomic molecule.
Combining
frequency v1 occurs, as for example in a heteronuclear diatomic molecule, the molecule will interact directly with infrared radiation of frequency v,. The intensity of absorption will depend upon (bg/bQl)02. Thus the "handle" by which the molecule interacts directly with light at the frequency of resonant ahsorption vl is the changing dipole m o m e ~ i t . ~I n order that Raman scattering can occur by interaction of the molecule with light at the much higher frequency UO, the molecular polarizability a must change during the vibration as discussed above, and the intensity of the Raman scattering will depend upon (ba/bQl)02. The "handle" for the interaction of the molecule with light in the Raman effect is the changing molecular polarizability. Since the intensity of Raman scattering and indeed the question of whether there is any scattering a t all depends upon the magnitude of the change in polarizability during the particular vibration, it is useful to consider a and changes in it during molecular vibrations in more detail. I n the discussion above, a was treated as a scalar, that is, as a simple proportionality constant relating the induced dipole moment to the inducing electrical field vector. Now the important quantity for a qualitative understanding of infrared absorption intensities, the dipole moment., is a vector involving three numbers p = p,i p,j p,k. I n general, it is easy to visualize a vector in three dimensions as an arrow. The polarizability, a, must relate one vector, the electrical field, to another vector, the induced dipole moment, where these two vectors are not necessarily parallel. I n general, a tensor which involves nine numbers is required for this, eqn. (9), and it is a 3 X 3 matrix. M = (10 E
+
Using the trigonometric identity cosaco@ ( C O S ( ~ 0) cos(a - P)] we obtain
+ +
=
'/a
The first term in eqn. (8) describes a classical dipole oscillating, and hence radiating, at the exciting frequency vo. The second and third terms represent "beat" frequencies of the light and molecular vibre tional frequencies and describe the Raman scattering a t frequencies (vo VI) and (vo - vl), respectively. For a polyatomic molecule with several normal modes of vibration, we can extend this simply by summing over the J different vibrational modes.
+
M
=
Eoao cos(2rvcJ)
+ '/&
J j-1
a(1
Qjo
(a)oX
Considering again the simple case of the diatomic molecule, the intensity of the light radiated a t uo, v V , and v - vl is proportional to the square of the time independent portions of the first, second and third terms, respectively, in eqn. (8) and to the fourth power of the frequency. This correctly predicts that the intensity of the Rayleigh scattering a t frequency vo depends on the equilibrium polarizability ao, while the Raman scattering depends on the derivative (ba/bQ~)o. One failure of the classical theory is that it predicts that the intensities of the Raman lines at vo UI and vo - n should be almost the same, while we noted vI has much lower inearlier that the line at vo tensity. Let us consider for a moment the factors leading to an infrared absorption spectrum. If the permanent dipole moment y of the molecule changes as vibrat,ion at
+
+
+
+
The physical significance of the tensor can be seen by considering, as examples, the elements in the second row. The diagonal element awuovv determines the magnitude of the y component of the induced molecular dipole moment which arises from the y component of the oscillating electrical field. The off-diagonal element ao, determines the contribution to the y component of the dipole caused by the x component of the electrical field, while ao,, determines the contribution to the y component of the dipole moment arising from the z component of the electrical field. To visualize the equilibrium molecular polarizability aO,we can draw arrows from a common origin which have lengths proportional to the value of ao in that particular direction. The heads of all of these arrows will define an ellipsoid. For some purposes it is more convenient to make the length of the arrows proportional to fro or to l/z/& We shall just use ao. Finally, we can make the polarizability matrix much simpler if we orient this ellipsoid with its principal axes along the x, y, and z axes of the coordinate system 4 Discussions of the interaction of infrared radiation and molecules can be found in Barrow's book (footnote 2) and in CRAWFORD, B., JR., "Chemieal Analysis by Infrared," Scientific American, October, 1953, Offprint No. 257, W. H. Freeman & Co., San Francisco, Calif.
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used to describe the polarizability. I n this case only the elements along the principal diagonal of a,, wzz orovv, and ao,,,, differ from zero. Figure 5 illustrates the polarizabihty ellipsoids lor several simple molecules all drawn to the same scale? According to the predictions above, the larger the polarizability ellipsoid the more intense the Rayleigh scattering at the exciting frequency, and this has been verzed experimentally. For a discussion of Raman intensities, however, we must examine the extent to which the polarizability changes as the molecule vibrates, that is, we must consider (bor/bQJ~or more particularly the derivatives of all 9 components of the polarizability tensor. Figure 6 illustrates the way in which the polarizability changes with the normal vibrational modes of the carbon dioxide molecule. Only the symmetric stretching vibration of Cop leads to a change in the polarizability. We, therefore, expect to find light vl but not with vo + scattered with frequency vo vz or vo pa. I n contrast, the dipole moment only changes during the antisymmetric stretching and the bending vibrations, so we would expect infrared absorption only at frequencies uz and vt. For the symmetric stretching vibration, the polarizability ellipsoid becomes larger and then smaller at the frequency of the vibration, that is, it "breathes" a t the vibrational frequency. Since the axes of the polarizability ellipsoid do not move and so remain aligned
with the x, y, and z axes used to describe the polarizability, the off diagonal elements of (9) all remain zero. For such totally symmetrical vibrations of a molecule, it is only the diagonal elements of the polarizability tensor which vary and lead to Raman scattering. If we consider more carefully the two vibrations which lead to a loss in symmetry of the molecule, we can see that the polarizability will not change during these vibrations. At the two extremes of a vibration, the vibrational coordinate Q will have the same absolute value but opposite signs. The shapes of the molecule will be identical at the two extremes. The vibration does not cause any rotation of the axes of the polarizability ellipsoid away from the fixed x, y, and z axes either. Since the polarizability is essentially a bond property, it will have identical values a' at the two extremes of the vibration. I n so far as eqn. (5) which neglects derivatives of higher order than the
*
:Polarizability data from STUART,H . A,, " M ~ l e k i i l i i t n & t ~ ~ ~ , ' ~ Springer, Berlin, 1034, p. 221.
Bendinq, v, Figure 5. scale.
Polarimbility ellipsoids for some simple molecules drawn to
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Figure 6. Polorizobility changes during the vibrations of carbon dioxide lexaggeroted).
first gives an adequate description of the variation in the molecular polarizability during the vibration, i t follows that
For this equality to hold, ( b ~ / b Q must ) ~ he zero, and hence or' = oro. The off diagonal elements remain zero, because the polarizability ellipsoid axes are not rotated by the vibration. The diagonal elements (actually, in general, the average value of the three) remain constant, because the vibration does not change the size of the ellipsoid. A somewhat more difficult case to visualize is that of an angular molecule like SOz. Figure 7 illustrates the changes in the polarizability during the three normal modes of this molecule. For the totally symmetric stretching vibration ul, the polarizability ellipsoid "breathes" with the vibrational frequency. With an angular molecule, the bending vibration does not change the symmetry of the molecule, and so the polarizability ellipsoid also periodically swells and contracts at the bending frequency up. In general for any totally symmetric vibration, the diagonal elements of the polarizability change, but the off diagonal elements remain zero. Raman scattering should, therefore,
be observed a t both vo + v l and vo + vz. Since the molecule has the same size and shape a t the extremes of the antisymmetric bond stretching vibration, the size of the polarizability ellipsoid does not change with time. The polarizability is an extremum in the x, y, and z directions when the molecule is in the equilibrium configuration with the ellipsoid axes aligned with x, y, and z; consequently the diagonal elements have their maximum or minimum values in the equilibrium configuration. This requires that the derivatives of the diagonal elements with the vibrational coordinate, ( b ~ / b Q ) all ~ , vanish. Although the ellipsoid does not change in size during this antisymmetric vibration, it rocks to and fro with frequency us as can be seen by reference to Figure 7. Since the axes of the polarizability ellipsoid oscillate with respect to the fixed x,y, and z axes during the vibration, the off diagonal elements of (9) change with time, and they are responsible for the Raman scattering at the frequencies uo us. The state of polarization of the Raman scattering yields valuable information concerning the molecular vibrations. If unpolarized light is used to excite a Raman spectrum, the scattering at 90" to the incident light will be found to be a t least partially polarized. The extent of the polarization depends upon the way in which the polarizability ellipsoid varies during the vibration, so polarization measurements provide additional information about the nature of the vibration. Experimentally, it is more convenient to use polarized incident light to excite the spectra. Let us consider the arrangement illustrated in Figure 8. The exciting light is directed along the x axis, and the
*
Symmetrical StretchinQ,v,
11. ~
~
Figure 7. Polarizability shonger during the vibrations of sulfur dioxide (exoggerdedl.
I. Totally
Symmetric Vibration
Non-Totally
Symmetric Vibration
Figure 8. State of polorimtion of the Roman scattering from totally symmetric and non-totally symmetric vibrations. Arrows indicate polorizotion in the plane of the poper, and circler indicate polorimtion normal to the plane of the poper.
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scattering is viewed a t a right angle to it along the y axis. The molecular vibration in Figure 8-1 is taken to be a totally symmetric one. The polarizability ellipsoid periodically expands and contracts, that is, only the diagonal elements are changing. The induced dipole will oscillate parallel to the inducing field direction. The sample is illuminated first with light polarized in the xz plane (a). The induced dipole oscillates in the xz plane, and plane polarized light is scattered in the y direction. Next the sample is illuminated with light polarized in the xy plane (b). Now no light is scattered in the y direction, because a dipole does not radiate along its line of action. The ratio of this scattered light intensity to the first is the depolarization ratio, p, and for this case it is zero. For molecules which have polarizability ellipsoids which are not spheres, the light scattered in the y direction in (b) will not vanish completely if all of the different orientations of the molecule in a liquid or solution are considered, but the scattered intensities in the two experiments will still differ ( p < 6/7). For a vibration which destroys some of the symmetry of the molecule, the intensity of the light scattered is almost the same in the two experiments ( p = 6/7). This situation is illustrated in Figure 8-11. Now the average value of the diagonal elements remains constant, while the off diagonal elements of the polarizability are changing; and these lead to rotation of the induced dipole away from the direction of the inducing field. Experimental determination of the depolarization ratio can usually provide a means of distinguishing symmet,ric from non-totally symmetric vibrations of the molecule. Summary
Raman scattering arises when a light wave sweeps an oscillating electrical field over a molecule and induces a transition of the molecule from a lower to a higher energy state or vice versa. I n order to have a vibrational Raman spectrum such as has been discussed, the molecular polarizability must change as the molecule vibrates. Thus the question of whether a particular type of vibration will lead to a Raman line depends upon whether the molecular polarizability varies or not during the vibration. The intensity of the line depends upon the magnitude of the polarizability change during the vibration. Thus the Raman spectrum yields information about both the shape of the polarizahility ellipsoid and the extent of its changes with the vibrations of the molecule; and these, in turn, give information about the molecular structure
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and the nature of the chemical bonds within the molecule. I n contrast, an infrared spectrum also allows deductions to he made about the molecular structure and bonding, but it provides information on the molecular dipole moment and the magnitude of its change during a vibration. I n studies of molecular structure and bonding, it is desirable to have both Raman and infrared spectra for the compound. Since the experiments involved in obtaining these as well as the fundamental basis for the two kinds of spectra are quite different, it is often possible to study a system by one technique but not by the other. Frequently it has bepn found to be difficult or even impossible to obtain infrared spectra of inorganic systems, and it is for this reason that Raman spectroscopy is again proving extremely valuable. Literature Cited ( 1 ) FERRARO, J. R., J. CHEM.EDLIC., 38, 201 (1961). ( 2 ) SMEKAL, A., Nalurwiss., 11, 873 (19'23). ( 3 ) KRMIERB, H. A., AND HEISENBERG, W., Z. Ph@., 31, 681 11925). (4) R.&&'c. V., Indian J. Phys., 2, 387 (1928). ( 5 ) RAMAN, C.V., AND KRISHNAN, K . S., Nature, (London), 121, 501 (1928). ( 6 ) LANDSBERG, C., AND MANDELSTAM, L., A T a f u m i s ~16, . , 557, 772 (1928). ( 7 ) RICE,F. O., AND TELLER, E., "Structure of Mattel;" John Wilev & Sons. Ino.. New York. 1949. n. 220. (8) EVANS;J. C., "infrared ~ p e c t r o s e o pand ~ Molecular Strncture," (Editor: DAVIES, M.), Elsevier, Amsterdam, 1963, p. 225. (9) KOHLRAUSCH, K. W. F., "Raman Spektren," Springer, Leipzig, 1943: Edwards Brothem, Ann Arbor, 1945. (10) HIBBEN, J . H., "The Raman Effect and Its Chemical Applications," Reinhold Publishing Corp., New York, N . Y., 1939. (11) BHAGAVANTAM, S., "Scattering of Light and the Raman Effect," Chemical Publishing Co., Inc., New Yurk, N. Y., 1942. (12) GWCKLER, G . , Rev. Modern Phys., 15, 111 (1943). ( 1 3 ) CLEVEL~ND, F. F., "Determination of Organic Structures by PhysicalMethods," (Editors: BRAUDE,E.A.,INDNACHOD, F. C.), Academic Press. Inc.. New Ymk. N . Y.. 1955. p. 231: ( 1 4 ) WILSON,M . K., "Determination of Organic Structures by Physical Methods," (Edilora: NICHOD, F. C., A X D PHILLIPS,W. D.), Vol. 2 ; Academic Press, Inc., New York, N. Y., 1962, p. 181. ( 1 5 ) MIZUSHIMA, S. I., "Handbuch der Physik," (Editor: FL~~GG S.),Vol. E, 26, Light and Matter II,Springer, BerlinGottingen-Heidelberg, 1958, p. 172. ( 1 6 ) BRANDM~~LLER, J., AND MOSEK,H., "Einfiihrung in die Ramanspektruskopie," 11. Steinkopff Yerlag, Darmstadt, 1962. ( 1 7 ) CABANNES, J . , A N D ROCARD, Y., J. Phv8ique Radium, 10, 52 (1929).
