1 ye. .... ‘.. ‘-\, ' .I k» u - A C. “1. A“. 7|- ~ ":4“ ?‘ ~.'-.I’\‘... A RAMAN STUDY OF CRYSTALL-INE ETHYLENES AND A DETERMINATION OF THEIR STRUCTURE THROUGH MODEL CALCULATIONS OF THE LATTICE REGION Thesis for the Degree Of Ph. 9; MICHIGAN STATE UNIVERSITY GLEN III R ELLIOTT ‘1972 e - ......... I" LIBRARY I Michigan State ‘ University This is to certify that the thesis entitled A RAMAN STUDY OF CRYSTALLINE ETHYLENES AND A DETERMINATION OF THEIR STRUCTURE THROUGH MODEL CALCULATIONS OF THE LATTICE REGION presented by Glenn R. Elliott has been accepted towards fulfillment of the requirements for Ph.D. degree in Chemistry gfizm Major professor Date August 10, 1972 0-7639 alum: ‘ HUM: & SUNS' I 800K BINDERY INC. LIBRARY BINDERS “I” main-ll ABSTRACT A RAMAN STUDY OF CRYSTALLINE ETHYLENES AND A DETERMINATION OF THEIR STRUCTURE THROUGH MODEL CALCULATIONS OF THE LATTICE REGION BY Glenn R. Elliott In solid ethylene there are two possible orientations of the two molecules per unit cell consistent with the 'crystal space symmetry. Extensive previous study of the crystal has offered no conclusive evidence as to which orientation is correct. Model calculations of the Raman lattice spectra for the two orientations were undertaken to evaluate which of the structures would more satisfactorily reproduce the experimental observations. The investigation gives strong evidence that the orientation with crystal space symmetry P121/n1 (Czh)' the so-called "b—structure", is correct. The development of a general theory for model calcula- tions of infrared and Raman lattice spectra has been com- pleted and is discussed in some detail, and its applicabil- ity to crystal structurecbtermination is indicated. The model calculations were tested on the Raman lattice spectrum of crystalline benzene, where the structure is well known. The results not only indicate that calculations based on the correct crystal structure can reproduce observed spectra, Glenn Robert Elliott but also that they can provide information useful for sym- metry assignments where polarization measurements cannot be obtained. Several sets of carbon-carbon, carbon-hydrogen and hydrogen-hydrogen interaction parameters were evaluated during this investigation. It was determined that Williams' parameter sets, most notably his most recent values, are the best available for treating atom-atom nonébonded inter- actions in solid ethylene. Raman spectra of the intra- and intermolecular regions of pure ethylene and ethylene-d4, and of the mixed crys- tals ethylene (4%) in ethylene-d4 and ethylene-d4 (4%) in ethylene, have been obtained. With the exception of the fundamaflal region of pure ethylene, these Spectra have not been reported previously. Assignments of all the factor group components and of the matrix-isolated bands, in addi- tion to several C-13 isotopic peaks, are given. A RAMAN STUDY OF CRYSTALLINE ETHYLENES AND A DETERMINATION OF THEIR STRUCTURE THROUGH MODEL CALCULATIONS OF THE LATTICE REGION BY L Glenn R. Elliott A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Chemistry 1972 G--- 78"??? {a JOHN STUART ORLEMANN July 22, 1945 to August 19, 1969 "They wrote in the old days that it is sweet and fitting to die for one's country. But in modern.war there is nothing sweet nor fitting in your dying." Ernest Hemingway Notes on the Next War ii .l .l ACKNOWLEDGMENTS I must first acknowledge my parents whose example has given me the dedication and perseverance that was necessary to finally complete my formal education. Their constant interest and encouragement is deeply appreciated. I also want to thank Dr. George Leroi for the interest and at times great patience with which he has viewed this research. The personal interest which he has shown for those in his research group has been important in creating a relaxed and enjoyable atmosphere for doing research. Gratefully acknowledged is the financial support ob- tained from Michigan State University and the Office of Naval Research. Deep appreciation also goes to the Department of Chemistry for my appointment as an Assistant Instructor. I want to thank Bonnie L. Marris for drawing the won- derful portrait on the preceding page. Finally, I wish to thank the countless people who have made my stay here a very memorable one. Special mention goes to my good friends Frank Chi, James Olson and John Shock, and particularly to Mrs. Naomi Hack for numerous diversions which have helped me maintain my sanity. iv CHAPTER I. II. III. IV. TABLE OF CONTENTS INTRODUCTION . . . . . . . . . . . . . . . . TIEORY O O O O O O O O O O O O O O O O O O O A. B. General . . . . . . . . . . . . . . . . Intermolecular Lattice Vibrations . . . Lattice Modes and Symmetry Coordinates Crystal Dynamics . . . . . . . . . . . Intensities of Lattice Vibrations . . . Infrared Lattice Mode Intensities . . Raman Lattice Mode Intensities . . . EXPERIMENTAL . O O O O O O O O O O O O O 0 O A. B. C. Spectrometer . . . . . . . . . . . . . . cryostat O O O O O O O O O O O O O O O 0 Sample Preparation . . . . . . . . . . . BENZENE . C C O O O O O O O C O O O O O O O A. B. C. D. Introduction . . . . . . . . . . . . . . Crystal Structure and Symmetry Considera- tions 0 O O O O O O O O O O O O O O O 0 Interaction Potential . . . . . . . . . Results and Discussion . . . . . . . . . ETHYIJENE . . O C O O I O O O O O O O O O O O A. B. C. D. E. Introduction . . . . . . . . . . . . . . Crystal Structure and Symmetry Considera- tions 0 O O O O C O O O O O O O O O O 0 Experimental Results . . . . . . . . . . Theoretical O O O O O O O O O O O O O 0 Theoretical Results . . . . . . . . . . REFERENCES 0 O O O O O O O O O O O O O O O O V Page 10 10 14 18 18 27 34 34 37 38 42 42 42 46 49 58 58 60 66 79 85 98 TABLE II. III. IV. V. VI. VII. VIII. IX. XII. XIII. XIV. LIST OF TABLES Page Lattice parameters and molecular orientations for crystalline benzene . . . . . . . . . . Correlation table for crystalline benzene . Internal symmetry coordinates for benzene . Atom-atom interaction potential parameters. Calculated and experimental frequencies for the librational lattice motions of benzene. Comparison of model Raman lattice frequencies and intensities with experimental values for crystalline benzene . . . . . . . . . . . . Lattice parameters and molecular orientations for the two possible crystal structures of SOlid ethYIene O O O O I O O O O O O O O 0 Correlation table for solid ethylene . . . Internal symmetry coordinates for the libra- tional lattice motions of ethylene . . . . Observed of solid frequencies and relative intensities and matrix isolated ethylene . . . frequencies and relative intensities and matrix isolated ethylene-d4 . Observed of solid Observed Raman lattice frequencies and calcu- lated frequency ratios for ethylene and ethylene-d4 . . . . . . . . . . . . . . . . Atom-atom interaction potentials used in calculating the lattice dynamics of crystal- line ethylene . . . . . . . . . . . . . . . Minimum energy rotations and static lattice energies for the two possible structures of solid ethylene . . . . . . . . . . . . . . vi 43 45 47 49 53 55 62 64 65 71 78 82 84 87 LIST OF TABLES (Continued). TABLE Page XV. Observed and calculated librational fre- quencies for solid ethylene . . . . . . . . 89 XVI. Molecular polarizability tensor components for ethYJ-ene O O C O O O O O O O O O O I O 90 XVII. Eigenvectors for Ag and B9 motions for both possible ethylene structures along with their corresponding observed frequencies . . . . 92 XVIII. Calculated and observed frequencies and rela- tive intensities and calculated mean amplitudes of libration for the two possible ethylene crystal structures . . . . . . . . . . . . 94 XIX. Observed and calculated frequencies for SOIid ethYIGDE-d4 . o I o o c o o 0 o o o o 97 XX. Calculated and observed frequencies and rela- tive intensities and calculated mean amplitudes of libration for the two possible ethylene-d4 crystal structures . . . . . . . . . . . . 97 vii FIGURE 1a. lb. 2. LIST OF FIGURES Page Spray-on cell for Malaker refrigerator. . . 39 Condensation cell for Malaker refrigerator. 39 Raman spectrum of crystalline benzene in the lattice region . . . . . . . . . . . . . . 51 Raman spectra of the intramolecular region of solid ethylene . . . . . . . . . . . . . . 68 Raman spectra of the internal vibrations of C2H4 (4%) iSOlated in C2134 o o o o o o o o 73 Raman spectra of the intramolecular region Of SOlid ethy1ene-d4 o o o o o o o o o o o 75 Raman spectra of the intramolecular region of matrix isolated ethylene-d4 (4%) in C2H4 77 Raman spectra of the lattice re ion of ethylene and ethylene-d4, at 30 K. The peak marked with an asterisk is a laser fluorescence line . . . . . . . . . . . . . . . . . . . 81 viii CHAPTER I INTRODUCTION In the past decade a great amount of work on the study of crystals by optical spectroscopy has been carried out. This work has been stimulated by the quantitative under- standing of crystal motions brought about by the extension of normal coordinate analysis from studies of free molecules to its current deep involvement in predicting and describing normal modes and frequencies of Optically active crystal motions. Much of this work has been done on molecular crystals, although experimental studies and calculations have also been made on ionic, covalent and.metallic crystals.1 The primary interest in molecular crystals is based on the simplicity of the form of the interaction. Molecular crys- tals do not display the strong metallic or covalent bonds or the long range electrostatic interactions of the other crystal types, and in many cases dipole and quadrupole interactions need not be considered. Molecular crystal motions can be divided into two classes. Intramolecular or internal motions can be described as the perturbation of gas phase molecular motions by weak Van der Waals, quadru- pole or dipole interactions between different molecules. These same weak interactions restrict the free rotational 1 2 and translational motions of gaseous molecules, giving rise to intermolecular or external motions. Initially molecule- molecule interatomic interactions directly available from second virial coefficients were used to form interatomic potential functions. Early papers have established a qualitative under- standing of crystal motions by a group theoretical treat- ment.2 For molecular crystals, group theory predicts the appearance of lattice motions corresponding to rotations and translations, and the splittings of fundamental motions consistent with the number of molecules per primitive unit cell and its symmetry termed factor group splitting. Also, in cases where gas phase motions show degeneracy, group theory predicts splittings due to site symmetry, site group splitting. In the infrared, many studies of intramolecular motions and less frequently of intermolecular motions have been made. Intermolecular and intramolecular motions have both been frequently studied with the Raman effect. Much of this work has been reviewed by O. Schnepp.3 Where the crystal structure has been known, the results have agreed with group theoretical predictions. In cases where the structure is not known, correlation of observed results with group theoretical predictions can narrow down the choice of possible crystal structures. The basic theory and the infrared work in the area have been reviewed by Vedder and Horniga and by Dows.a With the development of improved instrumentation, spectral features of crystals could be more thoroughly 3 characterized. The use of grating spectrometers and inter- ferometers in the far infrared region have permitted studies down to 20 cm-l. The use of laser light sources and iodine filters in the Raman have permitted shifts as low as 10 cm- for single crystals and 3 cm.1 for gas phase spectra to be observed. With the use of these more powerful light sources and more sensitive detectors, the increased signal- to-noise ratio permits resolution on the order of one half cm"1 and observation of extremely weak vibrational motions. Finally with the development of cryogenic instruments and techniques, the cumbersome and restrictive handling of cooled samples has been virtually eliminated. Thus, high quality Spectra have been obtained in the intramolecular and intermolecular regions for acetylene7 and anthracene and naphthalene8 in the Raman and ethylene9 in the infrared. Indeed, much more difficult systems have been studied Spec- trosc0pically with good results. These include matrix isolation studies of xenon dichloride10 in the Raman and of free radicals such as OF,11 C1012 and monobromomethyl13 in the infrared, as well as fluorescence studies of iodinel.4 Our present ability to obtain high quality spectra permits a full characterization in the case of crystal motions: their frequencies, intensities, splittings and in many cases their space group symmetry. This full characterization of the crystal motions now offers a base to test the theoretical developments over the past decade. These developments include the 4 calculations of the frequencies, intensities and Splittings which are observed by spectrosc0pic techniques. Shiman- ouchi, g£_3l,15 have applied the Wilson GF method16 to the calculation of optically active crystal vibrations and Walmsley and Pople17 have shown how to determine the inter- molecular force field from molecule-molecule (center—center) pair interactions. An extensive review of this field is given by Venkataraman and Sahni.1 Recently a more sophis- ticated calculation has been reported,18 the self-consis— tent phonon method, which includes anharmonicity and ampli- tude corrections allowing studies close to phase transitions. Discussion of the sources of intensity of the lattice vibra— tions is given by Schnepp19 for infrared absorption in— tensities and by Cahill20 for Raman scattering intensities. These are reviewed in detail by Richardson.21 Models used to calculate exciton splittings are discussed by Taddei and Giglio22 and by Blumenfeld, §£_§l,23 Most of the studies to date have been made on molecules which possess a center of symmetry. These molecules offer two major advantages. First, they usually crystallize in simple structures which possess a center of symmetry and have only two or four molecules per primitive unit cell. This simplifies the calculation of the intermolecular force field for external motions where the crystal structure is known, and in cases where only the space group is known it simplifies the calculation of the minimum energy orientation through crystal symmetry. Second, in the majority of these 5 cases, the molecules occupy inversion sites within the unit cells. Therefore, no mixing of librational and trans- lational lattice motions can occur for zone center crystal motions. Further, only the librational motions are Raman active and only the translational motions are infrared active. This permits the complete experimental and compu- tational analysis of librational and translational lattice modes to be carried out independently. A great deal of progress has been made in the calcula- tion of lattice modes. Early calculations on ethylene24 and on benzene and naphthalene25 indicated that the short range hydrogen-hydrogen and hydrogen-carbon repulsion forces were of primary importance in determining lattice frequencies. Recent calculations on ethane, ethylene and acetylene using short range repulsions and a diatomic ap- proximation have shown reasonable results for translational lattice motions.26 Calculations using diSpersion forces and quadrupole-quadrupole interactions along with the short range repulsive forces have been made on carbon dioxide,17 cyanogen27 and oxygen.28 For cyanogen and oxygen, a slight- ly better fit was obtained by excluding the quadrupole— quadrupole part from the interaction potential. This was not possible for carbon dioxide since a center-center inter- action was employed. Calculations by Shinoda and Enokido29 on carbon monoxide and by Goodings and Henkelman3° on nitro- gen have used correction terms and have invoked anharmonicity to explain the deviations from experimental data. Suzuki 6 and Schnepp31 have shown that anharmonicity can be very important for molecules smaller than carbon dioxide. The major conclusions drawn from these results are that the dispersion part of the interaction potential is important. the prOper choice of interaction potential parameters greatly improves calculations, good fits can be obtained without the inclusion of higher multipole moments, and anharmonicity can cause large shifts from harmonic frequen- cies for small molecules. In light of these advances in the theory of crystal lattice motions, we have undertaken a thorough investiga- tion of the possibility of structure determination by com- parison of model lattice spectra calculations and experi- mental studies in the intermolecular lattice region. Our investigation was made on Raman-active external motions since the method of calculation of Raman scattering inten- sities from molecules which possess an inversion center is more solidly founded than is the calculation of their infra- red absorption. The choice of crystals for this examina- tion was based on their symmetry. Only crystals where the molecules occupied sites of inversion symmetry were con- sidered in order to insure that there would be no mixing of translational and librational motions. Benzene was chosen to test the theory for several reasons. First it has been studied extensively and has been shown to obey group theor— etical predictions. Second it is easy to handle. Finally and most important, its crystal structure has been determined 7 with a high degree of accuracy at several temperatures with both X-ray diffraction methods and neutron scattering techniques. Benzene also meets the requirements of symmetry by having an inversion center and by crystallizing in a highly symmetric orthorhombic Space group with molecules on sites of inversion symmetry. The results of the calculations of the intermolecular lattice spectra of benzene indicate that these calculations can reproduce experimental lattice spectra of Raman active librational motions within the accuracy of the lattice parameters, orientations and potential functions. Having determined the accuracy of the calculations, we feel more confident in our crystal structure determination of ethylene. Ethylene was a natural choice to test the theory since its crystal structure is not known exactly. Packing considera- tions and static lattice energy calculations indicated two possible crystal structures for ethylene.24 These two crystal structures did not differ in space group so group theoretical predictions could not distinguish these possi- bilities. The two possible crystal structures differed only in the relative positions of the molecular planes of the two molecules per primitive unit cell. We have calculated the molecular orientations, static lattice energies, librational frequencies and relative in- tensities of the librational motions using several potential functions obtained by different techniques. These calcu- lations were made for each of the two possible crystal 8 structures of ethylene. The results of the calculations indicate that the relative intensities offer the best test of molecular structure, since they appear to be independent of the choice of interatomic potential functions. The re- sults of this study leave no doubt which of the two possible crystal structures is correct. They also further support the theory that crystal structures can be determined by comparison of experimental data with model spectra calcula- tions of the intermolecular lattice region. CHAPTER II THEORY A. General The vibrational spectra of the intermolecular region of molecular crystals are a source of much information about the forces acting between molecules and about the molecular orientations within crystals. Because of their periodic nature, crystals represent ideal systems with which to study intermolecular interactions. This can be done by comparing calculated properties of these intermolecular interactions such as molecular orientations, elastic con- stants, external lattice frequencies and fundamental split- tings with experimental data obtained by X—ray and neutron diffraction, compressibility measurements and infrared and Raman spectra of the inter- and intramolecular regions. The independent study of external lattice motions is justi- fied on the basis of the normally large energy separation between the intermolecular motions and intramolecular motions. Energies of intermolecular motions rarely exceed 200 cm-1, while the lowest skeletal vibrations of simple molecules are rarely less than 600 cm-1. Because of this, coupling between motions of the same symmetry in these two classes is negligible. Using the oriented gas approximation?2 9 10 in which solids are viewed as being composed of oriented molecules whose physical properties remain unchanged from the gas phase, independent of the perturbations of sur— rounding molecules, it is not difficult to calculate lat- tice mode intensities for the optically active librations and translations. Relative lattice mode intensities are highly orientation dependent. Therefore, calculated rela- tive intensities and frequencies may permit assignment of crystal structures in the absence of complete X-ray or neutron diffraction data. B. Intermolecular Lattice Vibrations Lattice Modes and Symmetry Coordinates21 Intermolecular lattice vibrations can be classified as two types. Translational lattice vibrations can be repre- sented as translations of one sublattice with respect to the others. This is best described as a shearing motion of the sublattices against each other. The translational dis- placement of the qth molecule in the pth cell is de— noted by ra(p/q) where a represents the three directions x,y,z. Rotational lattice vibrations can be represented as the rotations of the molecules of one sublattice about their centers of mass with reSpect to the rotations of the molecules on the other sublattices. The librational dis- h molecule in the pth cell is denoted placement of the qt by Aa(p/q) where a represents the three axes of rotation x,y,z. The instantaneous position and orientation of a 11 molecule are given by Ra(p/q) rgIP/qI + ra(p/q) and. (1) O Aa(p q) AaIp/q) + Nam/q) (2) respectively, where the superscript 0 indicates the equilibrium position or orientation. For systems with linear molecules, the redundancy condition is given by ZlAa(p/q)lz = z IA°

]2. (34) where N is the number of unit cells in the path, Qn is the normal coordinate of the transition from Y" to Y', and g, is the instantaneous dipole moment Of the crystal. The evaluation of the integrated absorption intensity for the normal coordinate, Qn' requires the evaluation Of the factor . (35) In the harmonic approximation, this factor vanishes except for a change Of one in the quantum number of the normal coordinate, Qn' and E.# O. The determination of the integrated intensity requires the determination Of the instantaneous dipole moment, E' The instantaneous dipole moment can be expanded in terms of the normal coordinates about their equilibrium positions to give I3 = i» Rim) + A} g (OLE/30pm; + (36) 20 neglecting higher order terms. The first sum represents the contribution of the permanent moments over all the molecules to the instantaneous crystal moment. This term will not be considered here since it does not apply to the systems to be discussed in this dissertation, crystals com- prised Of molecules with a center Of symmetry. Substitution Of equation (36) into equation (35) shows that the condi- tion for infrared intensity of a given normal coordinate, is Qn' = , (37) ul/BQ )0 75 0- N n is (a The first factor on the left is non-zero for one quantum transitions. In fact for translational lattice modes, it equals (1.68 x 10‘15/MV")1/2, and for librational lattice modes, it equals (16.8/IU")1/2 where M and I represent the molecular mass and the moment of inertia for the normal mode respectively.19 The second factor represents the sum Of the derivatives of the contributions to the instantaneous dipole moment Of the crystal by the molecules, 1, Qpi/Oon. The instantaneous dipole moment Of molecule 1, Q}, will in general consist Of three parts; the induced molecular moment caused by the local electric field, the permanent moment and the short range distortion moment.20 The dis- tortion moment is not expected to be very large for small amplitude (v = OI—> 1) vibrations at very low temperature. 21 It was found that in the pure rotational spectrum of nitro- gen35 and hydrogen36 gases, that the distortion moment is very small and decreases with temperature. Therefore, it will not be considered here. The induced moment consists Of dipole—induced moments and quadrupole-induced moments. For molecules possessing permanent dipole moments, the quadrupole-induced moments will generally be insignificant. Permanent moment derivatives predominate over induced moments for librational motions. However for translational motions, they will not produce a change in the total instan- taneous moment of the crystal since only spatial distribu- tions are changed. For crystals composed of non-polar molecules, the intensity must be wholly derived from in- duced molecular moments. Therefore the remainder of this section deals only with induced molecular moments. The induced dipole moment is dependent upon the elec- tric field at the perturbed site and the polarizability of the molecule on that site by the relation A = Q.§.- <33) By substituting equation (38) into the second factor on the right in equation (37), we obtain - . . . i (apf/aonIO = gj(a§}/aon)o + (agf/aQnIOED . (39) If the molecule is located on a centrosymmetric site, the i second term vanishes because ED vanishes. Also, for trans- lational motions, it vanishes because the spatial orienta- tional remains unchanged and (Egg/Oon)o vanishes. For 22 librational motions, it must be considered. However, since it is proportional to the anisotropy of the polarizability, it may be small. The effect Of both of the terms in equa- tion (39) will be discussed in terms of internal symmetry coordinates next. Since the normal coordinates can be related to in- ternal symmetry coordinates by an orthogonality transform, an, equation (39) becomes (agf/aon)o = fi an(8H?/Bsm)o and (40) i i E (Og'/OQn)o = % an g (ag,/asm)o = 2 L z [Oi(OEi/OS )0 (41) mnmi’z'v m + (ng/Osm)§:] . The sum over i is over all molecules, q, in all the unit cells, p. The first term in the brackets in equation (41) can be expanded in terms of internal coordinates by (aei/OsmIo g (BEi/BRa(p/q) I. (42‘ Z Z q P x (5Ra(p/q)/Bsm)o NW2 5 a when 120: (agi/an QIp/qn. (43) for translational motions, and similarly for librational motions 23 (agi/asmIO = N'l/zg g Tgfiq g (651/8 Aa(p/q))o~ (44) The field at molecule i is calculated to be the sum over the contributions from the neighboring molecules, j; j here represents all molecules (p/q), if = .3; §i(j)IR(j).A(j)] . (45) Equation (45) states that the electric field at site i is due to the contributions from all the surrounding molecules, j. Further it indicates that the contribution from each molecule, j, is dependent upon the distance from site i to molecule j, R(j), and the orientation of molecule j, IA(j). Substituting this into equation (43), we Obtain i _ -1/ . i (as /asm)o - N 2g g ngm'aqu (p/q)/5Ra(p/q))o + Tgwi (as? (p/q)/5Ra(i) )o I. (46) Because of the symmetric relationship between the internal ccordinates Ra(p/q) and Ra(i) of the same internal sym- metry coordinate, this sum vanishes unless Tm,aq and Tm,ai differ in sign or magnitude.19 Substituting equation (45) into equation (44), we note that in this case the derivatives with respect to (Aa(p/q) will be nonvanishing only if p/q denotes molecule j. Thus only one term of the sum over j remains giving 24 i = -1/ u 1 (BE /Bsm)o N 2%. 531‘qu EXDIBE (p/qI/B AO(p/q))o. (47) The second term in equation (41) will contribute when librational motions are infrared active, i.e. when the sites are non-centrosymmetric. Since it has only been evaluated for linear molecules, we will discuss it only in that con- text. The u v element of the polarizability tensor g} for molecule 1 is i _ . . auv - ai_éuv + BaaVKIAu(1)1\V(1)I (48) in which aav is the average polarizability, K is the anisotropy of the polarizability and ‘Au(i) and [\v(i) are the instantaneous direction cosines relating the molecular axis of molecule 1 to the u and v crystal axes. The . . i . derivative Of the auv element Wlth reSpect to a symmetry coordinate, Sm' becomes ' ’1 nu . (aafiv/asm), = 3N (2 aavIc[T;r"'uq/\3(i)+Tm'qu3(1)]o (49) The evaluation of the equations (46) and (47) re- quires an expression for the contribution from molecule q in unit cell p to the electric field at site i. In molecules which do not possess an intrinsic dipole moment, the major contribution to the infrared intensity for lat- tice motions will be from quadrupole-induced moments. For linear, non-polar molecules, the contribution to the elec- tric field in the a crystallographic direction at site 1 due to molecule (p/q) is given by 25 i -.- 2 _ - Ram/q} (39/2D‘H5dadz. Zdz. Aa da) (50) where 9 is the quadrupole moment, D is the absolute magnitude of the equilibrium position vector of molecule d d are 11'?- its direction cosines in the crystal-fixed coordinate sys- (P/Q) relative to the molecule on site i, dx' tem, d d ,, d its direction cosines in the coordinate x" y 2" system native to the molecule (p/q) and AX, Ay, A z are as defined above. Thus dz, = a31dX + aasz + a33dz = g [\ada (51) where the elements of the matrix 2‘ are defined as follows £‘E. 51 where {J is a vector in the molecule axis system and r, is a vector in the crystal system. Substitution of equa- tion (50) into equation (47) for example requires the rela- tion (5E;(P/q)/5I\B(P/q))o = (39/2D‘II10dad)dz.-2dz.5 -2d - . A E ob b o)Pq (52) where 6 represents a rotation axis parallel to one of the x, y, 2 molecule fixed axes. When a dipole moment exists, the contribution of the dipole-induced moment and the intrinsic moments to the infrared intensity must be considered. To consider the effect of dipole-induced moments, it is only necessary to introduce the relation for the contribution of the intrinsic 26 dipole moment on molecule (p/q) to the electric field at site i. This is given by i = 4 _ Ram/q) (M/D )( 3d,.da + Ag) (53) where M is the magnitude of the dipole moment. If the molecules possess an intrinsic dipole moment then libra- tional motions will be infrared active since they vary the orientation of the permanent molecular moments. The in- stantaneous component Of the dipole moment on molecule i, ul, along the a axis of the crystalebased system is i - o . . Ea - MlIAa (i) + Aa(l)], (54) in which /\:(i) is the equilibrium direction cosine and Aa(i) is the change in the direction cosine. Using equa- tions (7), (9) and (40), we have i _ '1/ .. i (at wen). - N 2?. an g: g Tm,aq EON onto/mags) Since g} depends only on A.a(p/q) where (p/q) refers to molecule 1, equation (55) reduces to i '1/ . .. i . (5p. /aQn)o N 32 L iTm'ang /a Aa(1))° (56) mnm = 2 II N gtznsz'aiL M. (57) Evaluation Of the intrinsic moment contribution to infrared intensity for librational motions is now possible by sub- stituting equation (57) into equation (37). 27 Raman Lattice Mode Intensities21 The total intensity Of Raman scattering associated with a transition from some state a to some state b is given by the quantum mechanical expression -1 Iab = NaC'(vi - ¥)4g|P|2(1 - exp(-hy{kT)) I (58) where Na is the number of molecules in the initial state, a, vi is the frequency Of the incident light, Vn is the frequency Of the Raman shift, 9 is the degeneracy Of vi, and C' is a constant.37 The induced transition moment, |P|, for a given normal coordinate, Qn' is given by IPI = IgIYbIon». (59) The field on a particular molecule is due to the electric field, g, of the incident radiation and the fields induced by the surrounding molecules. The induced moment, 2, then becomes 3 ' E gdlg ‘kgi Tik 2km ' (60) where %d is the polarizability of the perturbed molecule, 2: is an identity matrix, Tik is an element of the reac- tion field tensor, and %k is the polarizability Of the molecules surrounding the perturbed site, i. The effec- tive polarizability Of the crystal go can be expressed as 2&2'§%d [g'kii Tik %<]' (61) 28 In the infrared there were three factors which con- tributed to the integrated intensity; the distortion moment, the induced moment, and the permanent moment. As in the infrared, the distortion moment is not expected to contrib- ute appreciably to the Raman scattering. Also the perman- ent moment has no significance in the Raman. Therefore, the total scattering can be attributed to the induced moment. First, we will examine the effect Of the induced moment on the scattering from translational lattice vibra- tions. This is expected to be small because translational lattice vibrations do not change the orientation of the molecules within the crystal. Thus, they are not expected to change the polarizability Of the crystal, go’ to any great degree. Next, we will examine the effect the induced moment has on scattering from librational lattice vibrations. Finally, the form Of the induced moment for polycrystalline samples will be discussed. Neglecting the distortion moment, we examine the Raman scattering from translational lattice vibrations. For translational motions, the orientation of the molecules in the crystal and hence their polarizability remains unchanged. Therefore, the induced moment for these translational mo- tions is wholly dependent on how the reaction field tensor varies with these motions. The reaction field tensor varies with the normal coordinates, Qn = Qgcos(2wvnt), according to .. 0 o Tik - Tik + (aTik/aon)° on + ... , (62) 29 where Tik is the instantaneous reaction field tensor and Tik is the equilibrium value. The induced moment, g, of the crystal then becomes - o _ o . 5 ' 17‘ ERIE: kii Tikgac kii‘Mik/BQnMQn (213.1 Ex (”3) Including the form Of the normal coordinate given above and the time dependence Of the electric vector of the irradia- tive light, §'= E?cos(2wvit), we obtain g'- g gd[gfos(2wvit) -k§i Tikgk cos(2wvit) _ o kiiOTik/Oon)ocos(2wvnt)cos(2wvit)gk]E . (64) The terms in the induced moment which give rise to Raman scattering involve a change in frequency of (vi - vn) for Stokes scattering and of (vi + vn) for anti-Stokes scatter- ing. Since our interest is in the Stokes scattering, we will examine only the (vi - vn) frequency condition. With the relation 1 COS(2vvit)cos(2wvnt) I §{cos(27r(vi - vn)t) (65) + cos(21T(vi + vn)t)] and the condition that for scattering cos(27T(vi - vn)t) = 1, we Obtain = _ o o 13(Stokes) >13 93 z. (a'rik/aon)0 gkong /2. (66) k#i For centrosymmetric crystals, when (BTik/OQn)o is summed 30 over all the molecules in the infinite crystal, all Of its terms vanish due to symmetry and the induced moment for translational lattice vibrations is zero. Consequently, Raman scattering for translational lattice vibrations vanishes under the approximation Of infinite crystal and no distortion moment. This is not true for non-centrosym- metric crystals and therefore translational lattice motions can be Raman active in this case. In a similar fashion, the induced moment for librational lattice vibrations can be derived. For librational motions, the intermolecular spacing remains unchanged so the reac- tion field tensor remains constant. However, the molecular orientation changes with the normal coordinates according to On - Qgcos(2vvnt). The molecular polarizabilities can be written in terms Of the normal coordinates = a9 (Bad/aonuog + , (67) 2‘4 m With this relation and the time dependence of the normal coordinate and incident radiation, the induced moment be- comes : 0 - 0 _ ‘ 0 g 2.30.“; Z.Tikg3< “z Tik(5%i/don)°on l i¢k X cos(2wvnt)]§P cos(2wvit) O _ O " Res/awn. ... <2Tvnt>12 Z Tikgk k¢i -k:iTik(Ogd/Oon)003 cos(2wvnt)]EP cos(2wvit). (68) Since we are only interested in the terms which show 31 (vi - vn) frequency dependence, the induced moment reduces to : .. 0 O O P i gi kii Tik(5gk/OQn)an §,/2 - o o o + >; (a%/aon)otg >3_ Tikgkmne /2 (69) i k#i O o o = , - . 2. O (gagging 1.321 legglmong/ (7 ) Substituting equation (61) into equation (70) we Obtain the induced moment in terms of the crystal polarizability, gco g,= (Egg/OQn)oQg§P/2 . (71) This total induced moment for Raman scattering from libra- tions can be expanded in terms of individual components of the tensors and vectors 1/ = 0 2 2 2, Ion/2H2 [2(OapO/OQn)oEO] 1 , (72) p o where E0 is the component of the electric vector, E9, in the 0 direction and apo is the pO component Of the crystal polarizability tensor. The square of the induced transition moment defined in equation (59) is given by 1 (Plz = ||2- (73) Since only 0% is dependent on Ya(Qn) and Yb(Qn), 32 equation (73) can be written as 1 2 2 2 = _. o |p| 4 g [§(aapO/aon)oso] xl<‘1’a(Qn) IQnI‘l’b (on)>| . (74) In the harmonic oscillator approximation, the integral (Ra(on)log|wb(on)> for librations with a = O and b = 1 1 has the value (16.8/Ivnc) /3, where I is the moment Of inertia about the axis of rotation, vn is the frequency of the normal coordinate on, and c is the speed of light in centimeters per second. Substitution of equation (74) into equation (58) gives I U 2 Iab NaC (vi - vn)4g|l x [2 [2(aapO/Bon)oE012](4I1 - eXPI-hvn/kTII)‘1 (75> p O . K'(vi - Vn)‘9[% [g (OapO/OQA)OEO]2] x (vn[1 - exp(-hvn/kT)])-1 (76) where K' = (NaC'c x 16.8/4) and 08 is the mass weighted normal coordinate, Q; = 11/2Qn_ In our experimental system, we look at a random sample Of crystallites. This necessitates an evaluation of the last term in equation (76). By integrating over all pos- sible angles of incidence of the exciting radiation, we learn that the intensity can be evaluated with the assump- tion that one-third Of the radiation is incident from each of the crystal axes, X, Y, 2. Equation (76) is now evalu- ated for the three cases, Ex = 1/3. Ey = 1/3. and E2 = 1/3 33 I I I _ 4 ' . I 0 2 Iab K (vi vn) g %[ %(OOPO/don)EO] x (vn[1 - exp(-hvn/kT)]) 1 (77) = (1/3)K'(v. - v n)4g[z 2 up 02] p o x (vn[1 - exp(-hvn/kT)])'1, (78) where ago is the derivative Of the po polarizability tensor component with respect to the mass weighted normal coordinate, QB. In terms of the invariants 5' = (1/3)(a;{x + a§y + aéz) and (80) (7')2 = I1/2)I(a;(x - ayy)2 + (ayy - HQZIZ (81) +(a' zz - aéx)2 + 6[(d}'{y)2 + (a9z)2 + (a'zx)2]]. referred to respectively as the mean value and the anisotropy Of the derivative tensor, equation (78) reduces to k(vi - v n)4 g 2 Iab a Vn [1 ~1exp(éhvn Zkt)] [4 5(5 ) + (7 )2 1 (82) where K 3 (1/3)(3/2)K'. As we have shown, for centrosym- metric cells only librations show Raman scattering in the lattice region. Therefore, equation (82) is the only con- dition for Raman scattering intensity for polycrystalline samples in this region. CHAPTER III EXPERIMENTAL A. Spectrometer38 The spectrometer used for these experiments consists of a laser light source, a double monochromator, a photo- multiplier, and the associated electronics and Optics. For the laser light source, either a Spectra-Physics Model 140 Ar+ laser with a power output of approximately 1.5 watts each in the 5145 R and the 4880 2 lines or a Coherent Radia- tion Model 52BeO Ar+ laser with a power output of nearly 2.0 watts each in the 5145 R and the 4880 R lines was employed. Both laser sources have excitation line widths Observed to be less than 0.8 cm.1 at half height. The laser lines drOp sharply to the base line and permit the Observation of Raman scattering as close as 5 cm-1 to the exciting line for gas samples. In our analysis, the most highly scattering polycrystalline samples could be observed to within 10 R Of the exciting wavelength, permitting studies to within 40 cm‘1 of the exciting line for these cases. When using the 4880 A laser line at high power, a fluor- esence line at 4889 R appears with appreciable intensity. This caused little difficulty in either the intermolecular or intramolecular regions since all shifts were checked 34 35 with the 5145 2 laser line. No additional fluorescence lines were observed in any of the regions of interest, so broad band filters were not needed. The monochromator used is a Spex Model 1400. This double monochromator is, in effect, two Czerny-Turner 3/4- meter grating Spectrometers mounted in tandem. This is necessary to reduce stray light to a level acceptable for investigations of the weak Raman effect. Two sets of gratings were used. Initially, two Bausch and Lomb grat- ings with 1200 grooves/mm., blazed at 5000 X in the first order were used. A problem encountered with these gratings was the appearance of grating ghosts near the exciting wave- length, particularly when examining highly scattering poly- crystalline samples. This problem was somewhat alleviated by determining in advance the positions of the ghosts. Since they are symmetric with respect to wavelength about the exciting source, their positions and relative intensi- ties could be characterized by placing a scatter plate or ground glass joint in the sample area and focusing the scattered Rayleigh light on the spectrometer entrance slit. (Raman scattering, on the other hand, is symmetric in wave- number about the exciting line.) The subsequent purchase of a set of interferometrically ruled gratings from JobinéYvon eliminated the prOblem of grating ghosts. These select quality gratings were also ruled with 1200 grooves/mm. and blazed at 5000 X in the first order. The spectrometer has three slits which are 50 mm. high and can be bilaterally 36 adjusted from a few microns to 3 mm, and is equipped with a speed control which permits the selection of several scan speeds. The spectrometer was calibrated using argon, neon, and krypton gas discharge lamps. It was determined that a linear relation between dial reading and actual wavelength fit the calibration as well as any higher order polynomial function. Thus a constant correction factor was used in generating conversion tables from wavelength dial reading of the monochromator to wavenumber shifts. However, be- cause of the daily vagaries in the monochromator drive, it is necessary to obtain the correction constant by deter- mining the difference between the dial reading and the true wavelength of the laser excitation line before each experi- ment. For the detector, either a selected ITT FW-130 Star- Tracker end-on photomultiplier which has S-20 response or a selected RCA C31034 with similar characteristics along with slightly improved red response can be used. Both detectors are cooled thermoehxxrically to -20°C in a cooler-photo- multiplier housing assembly manufactured by Products for Research. When cooled to -20°C a dark cound of 3-4 counts per second can be achieved. A floating lens assembly pro— vided by Spex allows precise alignment of the radiation from the exit slit on the detector. With this assembly, good signal-to-noise ratios can be obtained down to the 10-11 ampere output range on our Victoreen Model VTE-l micro- microammeter. 37 B . Cryostat A Malaker Corporation Cryomite Mark VII—C was used to cool and control the temperature of all the solid samples. This instrument works on the principle of adiabatic expan- sion of helium gas, much like a standard refrigerator works with freon. It is capable of controlling temperatures of samples over a wide range from 250K up to room temperature. The temperature control is stable to well within one de- gree fluctuation. This control permits studies of spectral features, such as frequencies, band widths and intensities, as a function of temperature. It also has the advantage that it is small and can be moved from sample preparation areas to the sample region of our Spectrometer with ease. Complete characterization of the Raman spectra of solids requires polarized spectra of single crystal samples. However difficulty in sample handling and preparation pre- cludes the obtaining of single crystal samples in many situations. When this is the case, as it is for ethylene, it becomes necessary to study polycrystalline samples. Since selection rules are based on the assumption of an infinite crystal, crystallite size is important. If crystal- lite size is too large, then complete random interaction of polarized radiation with the sample,as assumed in the de- velopment in the previous chapter, may not occur. 0n the other hand, if the sample is highly powdered, then break- down of selection rules, broadening of spectral lines, and high Rayleigh scattering are possible. The temperature 38 control of the Cryomite helps circumvent these problems by permitting the annealling of powdered samples and complete shattering of samples obtained from the melt to give ac- ceptable crystallite sizes. For these two types of sample preparation, two cold head assemblies were designed for the the Cryomite. The first is a Spray-on head where the gas sample is allowed to diffuse into the cell, which is mounted on the cold block of the cryostat, and quickly solidifies on the copper face of the sample cell. This head is illus- trated in Figure 1a. The sample formed by this means is usually of a highly powdered form and needs to be annealled before good spectra can be obtained. The second head as- sembly, shown in Figure 1b, passes the gas through the cop- per mounting block of the sample cell where it is liquified before entering the cell. Once the sample cell is full, the liquid is rapidly frozen to insure adequate shattering to give a polycrystalline sample. Since, with this technique the crystallites are relatively large with reSpect to direct deposition, the sample suffers from none of the dis- advantages of the spray-on technique. However, it may suf— fer from insufficient shattering, giving rise to only par- tially oriented crystallites. Of the techniques, the second uses more sample requiring about 300 ml S.T-P. of gas, nearly twice that required by the first. C. Sample Preparation All samples used were initially obtained at high purity. Spectral Quality benzene (99+%) was obtained from Matheson, 39 Figure 1a. Spray-on cell for Malaker refrigerator. a. Copper base which is attached to cold finger. b. 1” Kovar seal c. Spray-on nozzle. d. 1/8" Kovar seal. e. 1/16" thindwalled stainless steel tubing. f. Silver soldered. Figure 1b. Condensation cell for Malaker refrigerator. a. Copper base which is attached to cold finger. b. 1/2" Kovar seal. c. Condensation chamber in center of cell. d. 1/16" thindwalled stainless steel tubing. e. Silver soldered. 40 Figure 1a. it: e Figure 1b. 41 Coleman and Bell and used without further purification. The benzene was drawn into the second head assembly, Figure 1b, by vacuum distillation and then rapidly frozen and taken to 30° K. The appearance of the sample and the spectrum of the lattice region both indicated sufficient shattering of the crystal to assure complete polycrystalline behavior. Research Grade C2H4 (99.98%) was obtained from The Matheson Company. The C2D4 (99+%) was obtained from Merck, Sharp and Dohme of Canada. Both of these samples were also used without further purification. Of the two sampling techniques used for the ethylene study, the spray- on technique was favored over the techniques of growing samples from the melt for two reasons. The first reason was the ease of handling of the sample by this method. Second, resolution was very good with both techniques and since mixed crystal work required the use of the gas deposi- tion technique to minimize the possibility of cluster forma- tion of the guest molecule, pure samples were also studied this way for comparative purposes. In no case were impuri- ties evident in the spectra. CHAPTER IV BENZENE A. Introduction Benzene lattice motions have been studied many times both experimentally39"5 and theoretically.25r43'45 Its crystal structure has been determined by X-ray diffraction at 270°K by Cox47 and by neutron diffraction at 218°K and 138°K by Bacon, 32 31,43 It is because the structure and dynamics of the crystal are so well defined that benzene represents a good crystalline system with which to examine the accuracy of the model calculations of torsional lattice spectra. Here we will use only one of the most recent sets of atom-atom interaction parameters determined by Williams.49 The evaluation of several sets of atom-atom interaction parameters is left for the next chapter. B. Crystal Structure and Symmetry Considerations There is only one known phase of solid benzene. It has orthorhombic symmetry and the crystal space group is Pbca (D;g). The crystal lattice parameters and molecular orientation for the two temperatures studied by Bacon, g£_al§3 are given in Table I. The listed direction cosines are between the x, y, 2 molecule fixed coordinates (where z 42 43 Table I. Lattice parameters and molecular orientations for crystalline benzene. Molecular Orientation Lattice 2700K x y z Constants a -0.2783 -0.6476 0.7094 7.460 b 0.9691 -0.1674 0.2238 9.666 c -0.0262 0.7434 0.6684 7.034 1380K x y z a -0.3233 -0.6551 0.6830 7.39 b 0.9457 -0.1978 0.2580 9.42 c -0.0339 0.7293 0.6834 6.81 44 is chosen perpendicular to the plane of the molecule, x lies along the carbon-hydrogen bond which is most closely aligned with the b crystallographic axis, and y is placed to complete an orthogonal right handed system) and the a,b,c crystallographic axes. In crystalline benzene there are four molecules per unit cell, located on inversion centers at the corner and face centers of the unit cell. These molecules are related to each other by a series of screw axes parallel to the crystal axes and glide planes perpendicular to these axes. The direction cosines refer to the molecule located at the origin, (0,0,0), and the set of all molecules equivalent to it by pure lattice translation. This set of equivalent molecules will be labeled set 1. The remaining molecules at (%u %u 0), (%u 0, %0, and (0, %, $0 and the associated sets of translationally equivalent molecules will be labeled sets 2, 3, and 4 respectively. The molecules of set 1 are related to those of sets 2, 3, and 4 by screw rotations parallel to the a, b, and c crystallographic axes respec- tively. Benzene has three rotational degrees of freedom. Since there are four molecules per unit cell, there will be twelve librational degrees of freedom. When these are classified in terms of the factor group Dzh’ it is found that there are three librational motions belonging to each , and B3 . of the irreducible representations Ag. B . B2 9 19 9 All twelve of these librational motions are permitted to show'activity in the Raman according to selection rules, 45 Table II. Correlation table for crystalline benzene Molecular Site Factor pOint group D6h group Ci group D2h Activ1ty Alg 2 2 2 (R2) A29 A9 x , y , z B B xy 1 1 9 Ag 9 Bzg——' B29 xz (RX.Ry) E1g Bag yz E29 A1u (T2) Azu Au inactive Biu Biu X Au Bzu Bzu y (TX.TY) Elu B3u z E 311 46 since they transform according to the polarizability tensor elements. The correlation table is shown in Table II along with the transformation properties of the factor group and the symmetry of the external motions of the D611 molecular point group. Symmetry coordinates were set up using linear combina- tions of molecule based internal rotational coordinates. These were right hand rotations about the molecule fixed axes, x,y,z. These are denoted by xa(£/k) where the sub- script a refers to the axis of rotation, x,y,z, and the coordinate refers to the kth molecule in the 2th unit cell. Since for the optically active lattice motions the mole- cules in all the unit cells are in phase, the internal co- ordinates are rewritten in the form hams) = N’1/2 >3 )xa(£/k) (33) where N is the number of unit cells in the crystal. Using the screw rotations, linear combinations of these coordinates may be constructed which have irreducible symmetry in the factor group. The symmetry coordinates are listed in Table III. C. Interatomic Potential In aromatic hydrocarbons there are three types of atom- atom interactions; carbon-carbon, carbon-hydrogen, and hydrogen-hydrogen. Each type displays an interaction po- tential of the form 47 Table III. Internal symmetry coordinates for benzene. Coordinate Symmetry 8.. = é-[xx<1> - xx(2>.- xx(3> + xx<4)1 Ag 51. = é- [5(1) + 5(2) - 5(3) - >3,(4)1 815 =§nz(1) - 5(2) + x203) - xz(4)1 Sm=%[kfl)+MW)-MQ)-MMH mg Sn=%[§H)-§@)-§W)+5MH $18 =%Hz(1) + xz(2) + )‘z(3) + xz(4)] [xx(1) - x (2) + x (3) - xx(4)] 1329 [x (1) + A (2) + xy(3) + xy(4)] m N o II hflH bflH~hflH [gu>+xm>-xm)-xmn Sm=%[MU)+&@)+MB)+&MN %9 s23 =%{>\1(1>- 5(2) + V3) - Vim 324 = é-[kz(1) + xz(2) - xz(3) - xz(4)] 48 )-D eXp(-C-Rij). (84) where Rij is the interatomic distance and A, B, C, and D are constants. If C = 0 and D = 12, then equation (84) reduces to the Lennard-Jones [6-12] potential function. For C ¥ 0 and D = 0, it becomes the Buckingham potential function. From among the many sets of atom-atom interaction potential parameters available in the literature, we have chosen one of Williams most recent sets.49 Many other sets of potential parameters will be tested in the next chapter. The atom-atom interaction parameters used for the benzene calculations are given in Table IV. Williams' parameters were derived by fitting the A and B constants in equa- tion (84) to permit agreement between observed and calculated crystalline prOperties such as structure, elasticity con- stants, compressibilities, and sublimation energies. The magnitude of the exponents were taken from a calculation of the interplanar spacing and compressability of graphite for c--°C and a quantum mechanical calculation of the repulsion of two hydrogen molecules for H---H. The C°"H eXponent was taken as the mean of these two values. The potential parameters have been modified and improved with the accumula- tion of more data since they were first published. It is because of this continued improvement and the amount of data to which these parameters have been fit that this set of potential constants was selected for the test of the model calculations of Raman lattice spectra. 49 Table IV. Atom-atom interatomic potential parameters. Interaction Parameters type A B C D H-H 1.74 1.704 3.74 0 C-H 9.68 7.703 3.67 0 C-C 53.90 74.620 3.60 0 A, units of 10-12 ergs 86; B, units of 10-10 ergs; C, units of 3-1. D. Results and Discussion A polycrystalline benzene sample was prepared by rapidly cooling the melt through the freezing point. The crystal was examined using the 5145 R argon ion laser line with a power output limited to approximately 100 milliwatts to minimize localized heating effects. The sample was main- tained at at approximately 400K. Spectrometer slits were set at 15 microns, giving a resolution of approximately 0.25 cm-l. All reported frequencies are accurate to within i1 cm_1. The spectra of the Raman lattice region of benzene is shown in Figure 2. Although twelve librational lattice modes are predicted according to selection rules, only six are observed in the present work. Ito and Shigeoka42 have measured the Raman spectrum of benzene and fully deuterated benzene at various temperatures down to that of liquid helium. They have 50 .&oov um coammu wofiuumH mnu Ca ocmucmn mcflaamummno mo Esuuommm cmEmm .N wusmflm 51 "Is. 0 -§ ‘} ~8 AllSNBlNI Figure 2. 52 observed six bands at liquid helium temperature. (Dows and coworkers have recently reported as many as eight,45 including two coincident bands.) In addition, Ito and Shigeoka have extrapolated values for Raman librational fre- quencies at 1380K. Since our calculated frequencies are based on the 138°K lattice constants and molecular orienta- tion, these extrapolated values permit a direct comparison between experimental and calculated librational frequencies. The calculated and experimental frequencies from this work and the experimental 4°K and extrapolated 138°K frequencies from Ito and Shigeoka are presented in Table V. The correla- tion between the extrapolated and the calculated frequencies in Table V is quite satisfactory The calculation of librational intensities was under- taken as described in Chapter II with one change. Since we assumed oriented gas behavior for crystalline benzene, the reaction field tensor was set equal to zero for the calcu- lation. For the evaluation of the derivatives of the ele- ments of the polarizability tensor with respect to the nor- mal coordinates, eigenvectors obtained from the frequency calculation were used. The values used for the molecular polarizability tensor components of benzene were not im- portant since only relative intensities are calculated, and these are dependent only upon the relative differences of the polarizability along axes perpendicular to the axis of rotation. Thus one can choose arbitrary values for the di- agonal elements of the molecular polarizability tensor, 53 Table V. Calculated and experimental frequencies for the librational lattice motions of benzene. Present Work Itoif’ Symmetry Calculated Experimental 4°K 138°K (1380K) (400K) Ag 95 98 (100) (90) 80 84 86 79 73 63 64 57 Blg 122 134 136 128 92 -- <86) <79) 77 -- 69 61 B29 107 -- (100) (90) 97 98 100 90 90 -- 86 79 B39 123 134 136 128 103 105 107 100 77 68 (86) (79) . . . -1 FrequenCies are given in cm . The experimental values in brackets are suggested frequencies for transitions which are forecast to be of low intensity in Frfihling's calculation, ref. 41. 54 restricted by the symmetry requirement that the polarizabil- ities along the two perpendicular axes in the molecular plane are the same and differ from the polarizability along the axis perpendicular to the molecular plane. The calcu- lated frequencies and their respective relative intensities are tabulated in Table VI along with their corresponding experimental values and the irreducible symmetries accord- ing to which they transform. The calculated relative intensities show good agreement with the experimental values within the restrictions of the calculation. First, the calculation was restricted to several temperatures where structural data are presently available. The observed spectrum to which the calculated spectrum was compared was obtained at approximately 400K. No structural data is available for crystalline benzene at this temperature. Small changes in the crystal structure could change the form of the eigenvectors and hence the intensities which correspond to them. Also, the inclusion of the reaction field tensor could change the form of the unit cell polarizability tensor. This would change the polarizability tensor derivatives and thus change the in- tensities corresponding to the various normal coordinates. The reaction field tensor was not included because oriented gas behavior was assumed. However Dunmur50 has shown that there is significant interaction of molecules within con- densed phases of several aromatic hydrocarbons, so as to alter the values of their molecular polarizabilities by as 55 Table VI. Comparison of model Raman lattice frequencies and intensities with experimental values for crystalline benzene. S mmetr Calculation (1380K) Experiment (400K) y y Frequencies Intensities Frequencies Intensities 33 123 .296 g (.597) 134 .74 B 122 .301 19 329 107 .010 B 103 .049 105 .03 39 B2 97 .017 g (.189) 98 .33 A 95 .172 9 B1g 92 .001 B 90 .044 29 Ag 80 1.000 84 1.00 B1 77 .007 9 (.057) 68 .03 B 77 .050 39 Ag 73 .254 63 .36 O O -1 I O O O C FrequenCies are in cm . All intenSities are given relative to the most intense band. Values in brackets indicate total calculated intensity which would fall within the experimental band width. 56 much as ten percent of their gas phase values. However, no data were given for benzene. Finally, in the crystal, benzene is slightly deformed from its regular hexagonal structure.47 Because of this, the low-lying E2g molec- ular vibration can mix with the lattice motions and change the form of eigenvectors. The degree of mixing induced by this deformation is not known, and therefore cannot be accounted for. The use of calculated relative intensities along with calculated frequencies allows the assignment of symmetries to bands observed in polycrystalline samples where polari- zation measurements are not possible. Our calculations offer a basis from which we are able to make corrections in the symmetry assignments of Ito and Shigeoka"2 and also in the predicted band intensities calculated by Friihling.41 Our results indicate that all the Ag modes show appreciable intensity and that none of the B39 modes are intensity forbidden in disagreement with Frfihling's calculations. Frfihling predicted the high frequency Ag mode and the low frequency Bag mode to have low intensity. His predictions were based on the assumption that these two bands, and two others which we calculate to be extremely weak are rotations purely about the axis perpendicular to the molecular plane. Our analysis indicates that this is not true for any of the librational motions. On the basis of Frfihling's calcula- tions Ito and Shigeoka assigned B1 symmetry to the band 9 which they observed at 69 cm-l. Our calculations indicated that this Big band is very weak and that the band 57 observed at 69 cm-1 by them and at 68 cm_1 by us has Bag symmetry. They also indicate that the majority of the intensity of the 98 cm"1 band arises from the Ag mode, not the Egg as Ito and Shigeoka suggest. In conclusion, model lattice calculations give a satis- factory account of the librational lattice motion of crys- talline benzene and also aid in assigning band symmetries. The atom—atom interaction parameters chosen do not appear to have caused any difficulties in the calculations. There is some theoretical justification for subdividing the re- pulsive interaction into a sum of atom-atom repulsions.51 However, the use of an atom-atom interaction to explain dispersive forces is questionable, especially for highly delocalized aromatic systems. Nevertheless, as a purely empirical function the atom-atom interaction is very useful. CHAPTER V ETHYLENE A. Introduction Within the past decade much work has been done on understanding interatomic potentials and their effect on the dynamic motions within crystals. During these years a large number of experimental and theoretical studies have been carried out on lattice vibrations of molecular solids. The initial thrust of much of this work was made with or- ganic crystals such as ethylene,24o52 benzene,25o53 hexa- methyl-tetramine,54 naphthalene,25:53v55 and antracene.54t55 In all these cases, except ethylene, the crystal structure was known and the solids were easy to handle. More recently interest has shifted to simpler molecular systems, most generally solids of linear molecules, in an attempt to get quantitative as well as qualitative agreement between ex- perimental data and theoretical calculations.3°:31 Of the organic systems, ethylene still stimulates a great deal of interest because it is one of the simplest non-linear sys- tems available. However, ethylene also poses a perplexing problem to lattice dynamicists. Lattice parameters and carbon positions 58 59 were determined in an X-ray investigation by Bunn,56 and Halford and Brecher57 using polarized infrared light to study a single crystal, determined solid ethylene to be monoclinic, with Space group ng. Dows24 subsequently pointed out that there are still two possible orientations of the two molecules in the primitive unit cell consistent with the C2; space group. In one case the two molecules are related by a screw axis parallel to the a crystallo- graphic axis, the "a-structure". In the other case they are related by a screw axis parallel to the b crystallo- graphic axis, the "b-structure". Much experimental and theoretical work has been done to determine the correct molecular orientation.22:23v24 These studies all indicate that the "b-structure" more satisfactorily fits the data. However, none offers a high degree of assurance that it is the correct structure. The present investigation utilizing Raman spectroscopy was initiated in an effort to solve the problem of the crystal structure, and to study the effect of the choice of intermolecular force fields on lattice dynamic calculations. Ethylene offers several advantages in regard to the second part: (i) Atom-atom interatomic potentials have been ex- tensively studied for carbon-hydrogen systems. For this reason, there is a wide choice of potential parameters de- termined by various methods to be examined.“3v53'63 (ii) The lowest frequency intramolecular motion appears at 810 cm.1 and therefore coupling between internal and external 60 motions is expected to be negligible. (iii) The crystal possesses an inversion center so that translational and librational lattice motions can be treated separately. B. Crystal Structure and Symmetry Considerations As with benzene, there is only one known phase of solid ethylene. This phase has been determined to be mono- clinic with either P21/n11 (ng) or P121/n1 (C22) space group. These two Space groups differ only in the direction of the principal axis of symmetry. We shall refer to them as the "a-structure" and the "b-structure" respectively, to indicate the direction of the two-fold screw-axis (and the normal to the glide plane) which is left in the resultant crystal. Using X-ray powder diffraction, Bunn56 determined the lattice parameters and the carbon positions as they ap- pear projected on the ab-plane at 980K. Bunn made the as- sumption of an orthorhombic structure with a Pnnm (Dzfiz) space group, based on the lattice parameters and carbon positions. Dows,24 using a purely repulsive potential, later corrected the structure by determining energy minima with respect to the hydrogen positions while maintaining Bunn's carbon positions. He found that there were two mini- mum energy orientations corresponding to rotations of the two molecules per unit cell about their carbon—carbon axes, first in the same direction and second in opposite direc- tions. An examination of the earlier investigations re- veals that all restricted the carbon-carbon bond to remain 61 in the ab—plane. We find that irrespective of potential function, the true energy minimum consistent with monoclinic symmetry is obtained only if the carbon-carbon bond is free to rotate out of the ab-plane, within the restriction of the symmetry elements of the Space group. Bunn did not ob- serve this rotation because he was only looking at the pro- jection on the ab-plane. The lattice parameters determined at 980K by Bunn and the molecular orientations determined by Bunn, by Dows (for the two monoclinic structures), and by us (for the two monoclinic structures) are reported in Table VII. The molecular orientations are defined in terms of direction cosines as explained in Chapter IV. Here x was chosen along the carbon-carbon bond, 2 was chosen per— pendicular to the molecular plane, and y was chosen to complete a right handed coordinate system. In crystalline ethylene there are two molecules per unit cell, located on inversion centers at the corners and center of the unit cell. These molecules are related by screw axes parallel to the a axis or the b axis and glide planes perpendicular to these axes for the "a—struc- ture" and the "b-structure" respectively. The direction cosines refer to the molecule at the origin, (0,0,0), and the set of molecules equivalent to it by pure lattice trans- lations. The remaining molecules at (%, én %) and transla- tionally equivalent to it form a second set of molecules. These two sets of molecules will be referred to as sets 1 and 2, reSpectively. 62 Table VII. Lattice parameters and molecular orientations for the two possible crystal structures of solid ethylene. Bunn56 Molecular Orientations Lattice 980K x y z Constantsa a .809 -0588 O 4.87 b .588 .809 0 6.46 C 0 0 1.0 4.14 Molecular Orientations "a-structure" "b-structure“ Dows24 x y z x y z a .809 -.525 .265 .809 -.515 .283 b .588 .722 -.365 .588 .709 -.389 c 0 .451 .892 0 .482 .876 1128:“ x y z x y z a .782 -.592 .194 .825 -.532 .190 b .614 .679 -.403 .542 .652 -.530 c .106 .434 .894 .158 .541 .826 aAll unit cell angles were observed to be 90°. 63 Ethylene has three rotational degrees of freedom, three translational degrees of freedom, and twelve internal degrees of freedom. Since there are two molecules per unit cell on Sites of inversion symmetry, only the intramolecular modes which are Raman active in the gas phase will be Raman active in the solid. Also, each of these Raman active in- ternal modes may appear as doublets consisting of an A9 and a B9 symmetry component. Among the external modes only the rotational degrees of freedom will be active in the Raman spectrum of the crystal. Since there are two molecules per unit cell we expect six Raman active libra- tional motions, three each of Ag and B9 symmetry. All the remaining intra- and intermolecular motions in the solid are infrared active. The correlation table for ethyl- ene is given in Table VIII. Symmetry coordinates were set up for the librational motions using linear combinations of the moleculeébased internal rotational coordinates. These were right handed rotations about the molecule-fixed axes, x,y,z. These are denoted by xa(£/k) where a = x,y,z and the coordinate refers to the kth molecule of the 3th unit cell. The zero wave vector coordinates are of the form x (k) = N’l/Z z A (z/k), (85) a 2 a where N is the number of unit cells in the crystal. The symmetry coordinates are listed in Table IX. Symmetry co- ordinates were not set up for the internal motions since Table VIII. 64 Correlation table for solid ethylene. Molecular Point Site Factor h Group Ci Group Czh Group D2 V1,V2,V3 stV6 V8 V4 v9,v10 ViiaViz (internal modes) (external mode) 1B 39 1Blu 31.1 19 3A9 18 3A < 29 g 39 9 3A 113 3A < u 211 u 1B 3Bu ’20 w IR(1-acous.) IR(Z-acous.) R and IR denote Raman and infrared activity respectively. 65 Table IX. Internal symmetry coordinates for the librational lattice motions of ethylene. Coordinate Symmetry "a-structure" s7 (2)-1/2[>\X(1) + xx(2)] Ag s. <2>"/2ny<1> - 51(2)] s. (2>"/2[xz<1> - xz<2>1 310 (2)-1/2HX(1) - xx(2)1 139 Sn (2)“/2ny(1) + 5(2)] s12 (2)-”nigh + xz<2n "b-structure" s7 <2>"/mx<1) - xx<2n Ag S. (2)-1/2My(1) + 3(2)] s. - (2)"/2[xz<1> - 42(2)] s1. <2)’1/2[x2<1) + xx(2)1 39 sh <2>“/2[xy<1) - xy<2>1 S12 (2)-1/2[7\z(1) + 02(2)] 66 no normal coordinate calculations were made on them. C. Experimental Results In all cases polycrystalline samples of either pure or mixed crystals were prepared by annealling the solid films formed upon gas deposition. The crystals, maintained at approximately 300K, were examined with the 4880 R and the 5145 R argon ion laser lines using a power output of approx- imately 1 watt. Spectrometer slit settings of from 15 microns (0.25 cm_1) to 150 microns (2.5 cm-l) were used. The reported frequencies are accurate to i 1 cm-l. The Raman spectra of polycrystalline ethylene and ethylene—d4 in both the intra- and intermolecular regions, and the mixed crystal spectra of 4% C2D4 in C2H4 and of 4% C2H4 in C2D4 at approximately 30°K were obtained. The spectral observations are entirely consistent with the C2; space group with two molecules per primitive unit cell located on sites of Ci symmetry. The Six Raman active modes in the gas phase, 3Ag, 2Blg, and 1B2g' give rise to six Raman active doublets, an A9 and a B9 component for each gas phase mode. Also there are six Raman active lat- tice modes, 3A9 and 3B9: librations corresponding to rotations in the gas phase. The factor group components of all six Raman active modes observed in the intramolecular region for C2H4 in the present study are depicted in Figure 3. The relative in- tensities are correct as shown with the exception of the v6 67 .ocwamcum oflaow mo cofimou HmHsomHOEmHucH map No muuowmm cmfimm .m oudmflm 68 .m muomflm onom 00.0..” L. (9.0m comm ..r 00.0— new. JEN..:(/\|JE 09.: man. 0mg 59 q q u q n is... N\ nmo mac 5? 69 mode at 1222 cm-1, which should be reduced by a factor of twenty. The v6 mode was not observed by Blumenfeld23 in the solid, probably because of its low intensity. It does agree with the weak absorption in the infrared assigned to it by Jacox.65 It was possible for Jacox to observe this absorption because of the highly powdered nature of her sample and the selection rule breakdown which accompanies it. Several of Blumenfeld's choices for factor group com- ponents have been corrected on the basis of our observed values. Also C-13 isotopic shifts were Observed for v2, v5, and v8. For v3, C-13 probably lies under the 1328 cm-1 component. The band at 1660 cm.1 is probably due to 2v10, This assignment is supported by the assignment by Halford and Brecher of the 828 cm-1 infrared absorption of crystal— line ethylene as the V10 ethylene motion.57 Five of the C2H4 fundamentals were observed in the C2D4 matrix. The V, band was not seen because of its low intensity. With the exception of the C-H stretching motions, all the factor group split components of C2H4 are very nearly split symmetrically about the isolated peaks. These modes also show very little in the way of site and factor group shifts from the gas phase values. The C-H stretching modes on the other hand show large site shifts. The matrix isolated spectra of CzH4 are shown in Figure 4. The additional peak at 3003.9 cm.1 has been assigned as an overtone of the C=C stretching motion of the C2D4 host lattice at 1508.8 cm-l, in Fermi resonance with the 70 isolated C2H4 v1 stretching mode. Such an assignment is unprecedented. However, the use of cubic terms in the anharmonicity calculation indicates that the overtone should be observed at 3002.: 2 cm-l. Further, such an overtone has the correct symmetry for interaction. Finally, the peak is not present in either of the pure crystal studies. The observed frequencies and relative intensities of the guest C2H4 in C2D4 and of the pure C2H4 solid state bands are reported in Table X. In C2D4, as for C2H4, all the factor group components of the Raman active modes have been observed in the intra- molecular region. See Figure 5. C-13 isotopic Shifts were observed for v1, v2, v5, and v8. Again the C-13 band for v3 is probably hidden under the 972 cm“1 component. The relative peak intensities and splittings are Similar to those observed in C2H4, although in all cases the split- tings of the factor group components in C2D4 are less than those in C2H4. The relative intensities are correct as shown in Figure 5 with the exception that the v2 peak has been reduced by one half. All the C2D4 peaks were observed when perdeuteroethyl- ene was isolated in C2H4 matrix at a mole ratio of 1:25. Within experimental error, all the factor group components are Split symmetrically about the isolated peaks shown in Figure 6. The site Shifts are small in all cases. C-13 isotopic shifts are seen for v1, v2, and v3. The experi— mental results for the C2D4 Spectra are listed in Table XI. Table X. Observed frequencies and relative intensities of solid and matrix isolated ethylene. Crystal Matrix Isolated I Imax cm-1 I Imax 2995.5 1596.7d .011 v2 (A ) c-C 1614.6 .386 1618.0 .199 9 1621.7 .003 . 1328.5 .578 1330.5 .022 ”3 (Ag) CH2 S°l° 1348.0 .234 1337.5 .719 a 3059.4d 009 v5 (B ) asym. C-H 3103 3065.9 '399 3068.0 .217 19 3068.3 ' 1222.4 .003 v6 (B19) CH2 rock. 1222 1226.7 .004 —-— -- 936.5d .008 941.6 .137 944.9 .154 V8 (Bzg) CH2 wag. 951.5 .020 aObserved frequencies taken from W. L. Smith and I. M. Mills, J. Chem. Phys. 49, 2095 (1964). bObserved frequency taken from B. L. Crawford, Jr., E. Lan- caster, and R. G. Inskeep, J. Chem. Phys. 21, 678 (1953). CObserved frequency taken from T. Feldman, J. Romanko, and H. L. Welsh, Can. J. Phys. 34, 737 (1956). dThese frequencies are attributed to C-13 isotopic shifts. 72 QVN GOSH owumaomfl Amvv vmno mo mcoflumunfl> accumucfl may mo manommm cmEmm .v musmfim 73 .v wusmflm nkom 000m. . no on meow nmo— o5— 9; ‘r~ - Ns JE J m) H). N\. 03: man— nno nmo t m> 74 .volmcwfihzuo Uflaom mo commou HMHSUOHOEMHUGA 0:» mo muuommm cmfimm .m musmflm 75 .m musmflm o—MN mmNN : OWNN 8mm . F 99 mm! a q J~ . 7 JM 4 4 , s m). H) N). ‘ . _ oak CNN 76 .vmuu Ca Amvv vvlwcmamsuw woumHomfl xfluuma mo coflmmu HmHsomHoamuucH mnu mo muuommm cmemm .o onamwm .m ousmflm 77 o—mm comm nnmu mmmm .p 9n— nav— up .1) . . as a E 4 Table XI. 78 solid and matrix isolated ethylene—d4. Observed frequencies and relative intensities of Gas Crystal Matrix Isolated cm-1 cm.1 I Imax cm-1 I/Imax 2236.1b .016 2237.2b .017 v1(A ) sym. C-H 2260 2244.5 9 2246.0 .687 2245.7 .723 1487.4b .022 1487 9 019 V2(Ag) 9‘9 1518 ig2§33 .857 1510.1 1.000 . 972.7 .352 v3(Ag) CH2 SCi 985 987.2 .197 980.9 .525 2290.6b .023 v5(B ) asym. C-H 2310 2302.0 2303.3 .455 1g 2305.5 1.000 996.9 .015 v6(319) CH2 rock. 1011 1002.1 .033 999.9 .040 b 773.7 .009 b v8(B2 ) CH2 wag. 785 778.7 .290 781.3 °gg§ 9 784.1 .030 ° ' aObserved frequencies taken from W. J. Lehman, J. Mol. Spectry. 1, 1 (1961). b These frequencies are attributed to C-13 isotopic shifts. 79 All the lattice modes have been observed for both ethylene and ethylene-d4. The relative intensities of the lattice modes are similar in the two cases. The ratio of the frequencies between corresponding bands in the two iso- topes indicates that all bands are due to librations, as was eXpected. Because of the small moment of inertia about the C-C axis, one would expect librations about this axis to have the largest frequency and to Show large Shifts on deuteration. They should also be weak because of the small difference in polarizability components perpendicular to the C-C axis. This is seen to be the case for the weak, high frequency pair of lattice bands although the fre- quency ratio is not 1.41 as would be expected for librations purely about the C-C axis. Figure 7 contains the spectra of the lattice region. The observed lattice frequencies for ethylene and ethylene-d4 and the calculated ratios of fre— quencies upon isotopic substitution are listed in Table XII. D. Theoretical For the calculations of the crystal lattice energy, lattice constants, and force constants, zeroth, first, and second derivatives of the potential with respect to the proper parameters were required. For these calculations, ten different sets of semi-empirical atom-atom potential functions of the type U.. = —A(R. 13 lj )-D eXP(-C-Ri.) (85) J 80 Figure 7. Raman spectra of the lattice region of ethylene and ethylene—d4, at 300K. The peak marked with an asterisk is a laser fluorescence line. 81 1 I ‘75 l l 160 VXu I I >=meb2~ 1 ISO CM" 1 l 100 Figure 7. 82 Table XII. Observed Raman lattice frequencies and calcu- lated frequency ratios for ethylene and ethylene-d4. Lattice Vibrations Ethylene Ethylene-d4 v v Vobs(cm-1) vobs(cm-1) C2H4/ C2D4 Raman 73 60 1.22 90 75 1.20 97 78 1.24 114 95 1.20 167 123 1.36 177 135 1.31 Infrareda - — _ 73 69.5 1.05 110 104 1.06 aInfrared frequencies obtained from M. Brith and A. Ron, J. Chem. Phys. 59, 3053 (1969). 83 were used. A, B, C, and D are selected constants and Rij is the distance between the i and j nonbonded atoms. As mentioned previously, for C = 0 and D = 12 the po- tential reduces to the Lennard-Jones [6—12] potential. For C # 0 and D = 0 it becomes the Buckingham potential. The values of A, B, C, and D chosen for calculations in this work for the hydrogen-hydrogen, carbon-hydrogen, and carbon-carbon nonbonded interactions are listed in Table XIII. The constants A, B, C, and D of the sets of potential functions I, II, III, and IV were obtained from hetero- geneous data on gas and solid phases of various molecules. See the references in Table XIII for more details. The constants of sets V through X were obtained by Williams by using a least-squares method applied to the results of a systematic investigation of several series of crystalline hydrocarbons. The least-squares method has been described in Chapter IV. Parameter sets V and VIII are derived from a least-squares fit applied to data on aromatic and non-aromatic hydrocarbons. Sets VI and IX are derived from the same combined data for the aromatic crystalline hydrocarbons, and sets VII and X are derived from data on non-aromatic crystalline hydrocarbons only. The calculations of the librational lattice frequencies, eigenvectors, and lattice parameters were made using the Wilson GF method16 as discussed in Chapter II. Summations were carried to six angstroms, giving approximately 80% of 84 00 0 00.0 0.00005 0.000 0 50.0 0.0000H 0.00H 0 V5.0 0.005H 5.00 X 0v 0 00.0 0.000V5 0.000 0 50.0 0.0055 0.00 0 v5.0 0.v05u v.5H XH 00 0 00.0 0.005H5 0.0H0 0 50.0 0.0000 0.HHH 0 v5.0 0.H5H0 v.00 HHH> 00 0 00.0 0.000H0 0.000 0 50.0 0.000ufi 0.00H 0 V5.0 0.0000 0.00 HH> 00 0 00.0 0.00vv5 0.000 0 50.0 0.0va 0.00H 0 V5.0 0.000v 0.00 H> 00 0 00.0 0.00000 0.000 0 50.0 0.0550 0.00H 0 v5.0 0.0000 0.50 > fi0 0 00.0 0.00000 0.000 0 0H.v 0.0000v 0.¢0H 0 00.v 0.0000v 0.50 >H 00 0H 0 0.000H00 0.500 0 v0.0 0.0H5vv 0.¢0fi 0 00.0 0.0000 00.0w HHH 00 0 00.0 0.000500 0.500 0 00.v 0.000H0 H.HOH 0 00.v 0.0000 00.00 HH 00 0a 0 0.000000 0.050 0H 0 0.00000 0.00H 0H 0 0.0000n 0.H0 H D U m < D U m d D U m 0 .000 010 01m Elm umm .mcwamnum mafiaamummuo 0o moflemcmv mofluuma 0:0 mcflumasoamo CH wows mHmHucmuom cofluomumucfi EouMIEou< .HHHN wHQmB 85 the static lattice energy.62 Minimum energy orientations and force constants were calculated for each of the two pos- sible structures with each of the ten sets of potential functions. For each of these cases a set of frequencies, eigenvectors, and mean amplitudes of libration was calcu- lated. Further, these eigenvectors were used to calculate the relative intensities of these lattice modes according to the method described in Chapter II. The oriented gas model assumption was made. Therefore the reaction field tensor was not included in the calculations of the polariz— ability element derivatives. E. Theoretical Results The energy was minimized with respect to the molecular orientation by allowing the two molecules per unit cell to rotate within the restrictions imposed by the symmetry of the space group. The final orientation of the molecules is explained in terms of rotations of the molecules from a position with the molecular plane in the ac-plane and the carbon-carbon bond parallel to the a crystallographic axis. The three angles of rotation, ¢ , B , and 9, taken in that order, represent right-handed rotations about axes parallel to the a, b, and c crystallographic axes. Space group symmetry requires that both the B and 6 rotations for the two molecules in the unit cell be equal in magnitude but opposite in Sign and the 0 rotations be equal in sign and magnitude for the "a-structure". For the "b—Structure" 86 the 0 and 9 rotations are equal in Sign but opposite in magnitude, while the B rotations are equal in Sign and magnitude. The list of rotations corresponding to energy minima for the ten sets of potential functions, along with the calculated static lattice energies, are given in Table XIV. Only the rotation angles for set 1 molecules are given. These rotations can be correlated with the molecular orienta- tions given in Table VII by substituting the angles of rotation into the orientation matrix cos Bcose cos esin Bsin 0 sin Bcos ¢cos 9 -cos ¢sin 9 +sin ¢sin 9 cos Bsine Sin ¢sin Bsin 9 cos ¢sin Bsin 6 (86) +cos ¢cos 6 -sin ¢sin 6 -sin B sin ¢COS B cos 0cos B . The variations of the ¢, B, and 9 rotations with the choice of potential set are greater for the "a-structure" than for the "b-structure", (in degrees, 2.6 gs. 1.0, 18.6 gs, 1.2, and 2.2 Kg, 1.8 respectively). This indicates a greater anisotropy of force on the "a—structure" unit cell than on the "b-structure" unit cell. More important is the fact that regardless of the choice of potential function, the calculated energy for the "b-Structure" is less than that of the "a-structure". However this only offers a qualitative indication that the "b-structure" is better. Knowledge of the frequency behavior of the zone center lat- tice librations and their relative intensities is needed to 87 Table XIV. Minimum energy rotations and static lattice energies for the two possible structures of solid ethylene. Minimum Energy Potential Struc- Orientations Energya Set ture m B 9 (Kcal/Mole) I a 26.1 12.5 40.1 -4.441 II a 24.9 12.1 39.7 -4.021 III a 25.9 10.3 38.1 -3.833 IV a 25.1 12.1 38.5 -4.816 V a 26.1 -3.9 40.3 -3.398 VI a 24.3 10.5 38.3 -3.535 VII a 26.9 -3.9 40.1 -3 .290 VIII a 26.5 -3.9 40.3 -3.070' IX a 26.7 -3 .9 40.1 -2.823 X a 25.9 -6.1 38.1 -2.963 I b 33.6 -9.1 32.5 -4.491 II b 33.0 -8.7 31.9 -4.124 III b 33.2 -9.1 32.1 -3.950 IV b 34.0 -9.7 33.3 -4.885 V b 33.4 -8.7 32.3 -3.664 VI b 32 .2 -8.5 31.5 -3.813 VII b 33.8 -9.1 21.7 -3.546 VIII b 33.4 -8.9 32.5 -3.284 IX b 33.6 -8.9 32.9 -2.998 X b 33.2 -9.1 33.3 -3.149 aStatic lattice energies are determined by summing atom-atom interactions out to six angstroms. Williams has shown that this gives a value of approximately 80% of the true value.62 For comparison, 80% of the sum of the latent heats of fusion and vaporization (-4.037) is -3.230 Kcal/mole. 88 provide a more sensitive test, since higher derivatives are utilized. The static crystal energy was also minimized with re- spect to the crystal unit cell dimensions. The small variations of the unit cell dimensions for the "b-structure" case indicate that the lattice parameters Obtained at 980K should be substantially independent of temperature. Further, this indicates that the lattice frequencies should show little variation with temperature. Thus a comparison of frequencies observed at 30°K and frequencies calculated using the 980K unit cell parameters should be a reliable method of determining which of the potential set is best. The frequencies calculated using each of the sets of interatomic potentials are listed in Table XV, along with the Observed values. The use of interatomic parameter sets I, II, III, and IV to calculate the force constants for librational motions lead to imaginary frequencies in sever- al cases. Their failure in calculating librational fre- quencies is due to their "softness“. They overemphasize the role of the attractive terms at small intermolecular distances. For this reason, they will not be considered further. The calculations using the remaining sets of interatomic parameters give frequencies which vary consider- ably in their ability to fit the observed values. The "b- structure" frequencies Show a Slight improvement over the "a-structure" frequencies in reproducing the observed fre- quencies. However, the improvement is not good enough to 89 Table XV. Observed and calculated librational frequencies for solid ethylene. Struc- - -1 Set ture Calculated FrequenCies (cm ) a 88 94 99 111 239 245 I b imag 21 52 68 116 137 a 97 102 105 117 236 247 II b (several imaginary frequencies) a 96 101 111 133 247 263 III b 37 68 96 99 164 173 a 105 110 125 135 259 269 IV b imag 61 96 97 130 155 a 75 97 101 158 252 260 V b 81 98 120 128 200 202 a 142 144 148 174 316 337 VI b 97 116 144 150 249 252 a 80 103 106 162 247 257 VII b 88 101 126 134 200 200 a 72 91 94 145 226 234 VIII b 77 91 111 120 181 182 a 70 87 89 136 206 214 IX b 75 87 105 114 167 169 a 79 94 99 146 200 224 X b 82 90 110 124 173 173 Obs. 73 90 97 114 167 177 90 offer any assurance that the "b-structure" is correct. The relative intensities of Raman scattering from the librational motions were calculated using the eigenvectors obtained from the frequency calculation and the diagonal components of the molecular polarizability tensor calculated by J. F. Harrison66 using uncoupled Hartree-Fock perturba- tion theory. A description of the quantum mechanical calculation has been published.67 The diagonal components of the tensor are given in Table XVI. Table XVI. Molecular polarizability tensor components for ethylene. Tensor a Component Value QXX 9.387 4.894 0£in 322 4.842 aAll val es were calculated by Harrison66 and have the units ( )3. For the discussion of the results of the intensity calculation, we will refer only to values calculated using the eigenvectors generating from set X potential parameters. This can be done without loss of generality because the in- tensity calculations are more or less independent of the choice of potential function. The calculation of relative librational intensities is dependent on three factors: (1) 91 the molecular polarizability tensor components, (2) the molecular orientations, and (3) the eigenvectors. Since the molecular polarizability tensor will remain unchanged, it will not effect a difference in the relative intensities calculated with different potential functions. The varia- tion of the molecular orientations with the choice of potential function is quite small and for the "b-structure" is effectively negligible as can be seen in Table XIV. Thus it will have very little if any effect on the calcu- lated relative intensities Obtained with different potential functions. Finally, the eigenvectors show a high degree of reproducibility, irrespective of the choice of potential function. With the exception of the first four sets of interaction parameters, which do not consistently produce real frequencies, the eigenvectors for the normal modes ob- tained using the various potential functions are Shown in Table XVII. Variations are apparent. However, large devia- tions do not exist. It appears that for a self-consistent set of interatomic potential parameters, eigenvectors, like structure, show a high degree of stability. The calculated frequencies, relative intensities, and mean amplitudes of libration are given in Table XVIII. Goodings30 has shown that a librational amplitude of 26 de- grees in solid nitrogen gives an anharmonicity of approxi- mately seventy percent. However, we feel that librations with amplitudes of less than ten degrees should Show an acceptable low anharmonicity for our calculations. The Table XVII. Eigenvectors for A 92 and B 9 motions for both possible ethylene structures, along with their corresponding observed frequencies. AgMotions Frequency Bg Motions 87 S10 S12 58 S9 S11 "a—structure" .535 -0024 -0012 .532 0027 0024 .538 .007 -.014 177 .533 .033 -.008 .535 .024 ‘0015 .532 .029 .024 .535 -.024 -.014 .532 .029 .025 .535 .024 -.015 167 .532 .029 .024 .534 .027 -.018 .535 .022 .016 0063 .198 .105 -0083 .124 .199 -.030 .197 -.114 97 -.019 .099 .218 .068 .199 .104 -.084 .129 .194 .066 .196 0110 -0085 .126 .197 .067 .193 .116 114 -.084 .125 .198 .076 .204 .090 -.060 .124 .201 .001 .097 .219 .024 -.182 .139 .020 .102 .216 73 .080 -.196 .108 .006 -.096 .219 .026 -.178 .145 .001 -.101 .217 .026 -.181 .141 .002 -.107 .214 90 .027 -.181 .140 .013 -.084 .226 .024 -.183 .138 Cont. 93 Table XVII. (Continued) Ag Motions Frequency B Motions 58 S9 S12 S7 810 S11 "b-structure" .537 -0019 “”0005 .539 -0004 -0004 .537 -.019 -.003 167 .539 -.004 -.002 .536 -0021 -0010 .539 -.004 -0009 .537 -0020 -0008 .539 -0004 -0007 .536 -.018 -.018 .538 -.005 -.018 0046 0221 -0012 0011 0053 .236 .046 .221 -.015 97 .007 .054 .236 .049 .220 -.021 .022 .055 .236 0047 0221 “.017 .018 .053 .236 .044 .221 -.013 114 .020 .054 .236 .039 .220 ‘0024 .041 .053 .236 .014 .010 .243 -.007 -.215 .058 .083 .014 .243 9° -.008 -.215 .059 .027 .019 .242 -.005 -.215 .060 0022 .015 0243 -0007 -0216 .059 .026 .011 .243 73 -.005 -.215 .060 .045 .020 .242 -.003 -.216 .059 These eigenvectors were generated from the last six sets of interaction parameters listed in Table XIII. The frequencies indicate the observed frequency to which each eigenvector belongs. 94 Table XVIII. Calculated and Observed frequencies and rela- tive intensities and calculated mean amplitudes of libration for the two possible ethylene crystal structures. Frequency (Amplitude) Relative Intensity Obs. Calc. "a" Calc. "b" Obs. Calc. Calc. Ila N "b N 177 224(10.9) 173(12.4) .002 .006 .001 167 200(11.5) 173(12.4) .010 .001 .006 114 146(5.6) 124(6.1) .197 .057 .177 97 99(7.6) 110(6.5) .418 .535 .628 90 94(7.2) 90(8.1) 1.000 .619 1.000 73 79(8.8) 82(8.6) .190 1.000 .314 I 111'“, I'll-I'll: 95 comparison of the observed values with the calculated fre- quencies and their relative intensities for both structures gives strong support to the "b-Structure". The inability of the "a-structure" calculations to predict the correct ordering of the intensities, along with the prediction that one of the strong low frequency librations Should be in- tensity forbidden, indicate that the "a-structure" cannot be the correct structure. In addition the ordering and magnitudes of the calculated "b-Structure" intensites are no more than fifteen percent in error. This is completely within the limits of accuracy expected. The limitations on the accuracy of the relative intensities include the accur- acy with which the molecular polarizability tensor is known, as well as the several approximations used in classical lattice dynamical calculations. Presently there are no ex- perimental values for the molecular polarizability compo- nents with which to compare the theoretical values used. Also, Dunmur5° has shown that perturbations do indeed exist in solids and can alter gas phase molecular polarizabili- ties by as much as ten percent. These perturbations are neglected in our oriented gas approach. Finally, calcula- tions of eigenvectors may be as much as five percent in error due to the form of the interaction potential function. Similar results have been obtained for ethylene-d4. In this case the electronic structure of the molecule and of the crystal will not be changed from the ethylene case results. Therefore the molecular orientations and static 96 lattice energies will be the same as those reported in Table XIV. Only potential sets VIII, IX, and K were used to cal- culate the ethylene—d4 frequencies, which are compared to the observed values in Table XIX. As was noted for ethylene, the "b-Structure" frequencies are in slightly better agree— ment with the observed values than are the "a—structure" fre— quencies. However, the improvement is not Significant. Rel- ative intensities were caluclated as for ethylene. The cal- culations were based on the eigenvectors generated with set IX interaction parameters, and the results compared to ex- periment in Table XX. Again the calculations for the "b- structure" provide strong evidence that this molecular orien- tation represents the correct crystal structure. The "a- structure" calculations cannot predict the proper ordering or relative intensities of the observed spectra. The con- clusions drawn from the ethylene-d4 calculations agree with those drawn from the ethylene calculations, that the "b- structure" must be the correct one. In conclusion, the calculations at all levels indicate the "b-structure" to be correct. Within the limitations of the model, the intensity calculations indicate with a high degree of assurance that the "a—structure" cannot satisfac- torily predict the Observed spectra. Further we have indi- cated that intensity calculations offer a powerful tool for the determination not only of symmetry assignments, but in- deed of crystal structure where an ambiguity remains from diffraction experiments. This is particularly significant since, unlike energy and frequency calculations, the rela- tive intensity predictions are effectively independent of the choice of intermolecular potential. 97 Table XIX. Observed and calculated frequencies for solid ethylene-d4. Struc- . -1 Set ture Calculated FrequenCies (cm ) a 61 75 78 122 161 166 VIII b 63 77 91 102 128 129 a 59 72 74 114 147 152 IX b 62 74 86 98 118 119 5 i a 67 77 82 122 144 159 i X _ b 70 80 92 108 124 124 9 !. Obs. 60 75 78 95 123 135 Table XX. Calculated and Observed frequencies and relative intensities and calculated mean amplitudes of libration for the two possible ethylene-d4 crys- tal structures. Frequency (Amplitude) Relative Intensity Obs. Calc. "a" Calc. "b" Obs. Calc. Calc. "a" "b. 135 152(9.4) 119(10.7) .013 .011 .001 123 147(9.6) 118(10.8) .082 .003 .010 95 114(5.3) 98(5.8) .198 .027 ‘.195 78 74(7.9) 86(6.2) .504 .621 .609 75 72(7.0) 74(7.9) 1.000 .951 1.000 60 59(9.2) 62(9.0) .170 1.000 .316 REFERENCES 11. 12. 13. 14. REFERENCES G. Venkataraman and V. C. Sahni, Rev. Mod. Phys. 42, 409 (1970). D. F. Hornig, J. Chem. Phys. 16,1063 (1948), H. Winston and R. S. Halford, J. Chem. Phys. 17, 607 (1949). O. SchnePP. 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