THE U N I V E R S IT Y OF M I C H I G A N COLLEGE OF ENGINEERING Department of Nuclear Engineering Technical Report APPLICATIONS OF THE DIRECTIONAL PHONON FREQUENCY FUNCTIONS IN NEUTRON SCATTERING INVESTIGATIONS OF HIGH POLYMERS Jcia -E. Lynch, Jr. ORA Project 08964 supported by: NATIONAL SCIENCE FOUNDATION GRANT NO. GK-1709 WASHINGTON, D.C. administered through: OFFICE OF RESEARCH ADMINISTRATION ANN ARBOR January 1968

This report was also a dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in The University of Michigan, 1968.

TABLE OF CONTENTS Page LIST OF TABLES v LIST OF FIGURES vi ABSTRACT x Chapter 1. INTRODUCTION 1 20 NEUTRON SCATTERING THEORY 3 A, Incoherent Scattering Cross-Section 3 Bo One- and Two-Phonon Directional Frequency Functions 7 C, Orientation Averages 1.1 D. Incoherent Approximation for Coherent Scattering Cross-Section.-9 3. LATTICE DYNAMICS 23 A. Dynamical Matrix Equation 23 B. Cartesia.n Symmetry Coordinate Transformation 27 C, Cartesian Force Consta.nt Matrices 3.1 D, Phase Dependent Matrix Equations 35 4. SINGLE-CHAIN POLYETHYLENE 37 A. Cha.in Structure and Coordinates 37 B. Intramolecular Force Constants 4.1 Ce Line Group Symmetry Analysis 43 D. Cartesian Symmetry Coordinates 47 E, Phase Dependent Dynamical Matrices 52 F. Dispersion Curves and One-Phonon Frequency Functions 60 5. CRYSTALLINE POLYETHYLENE 66 A. Crystal Structure and Coordinates 66 B Intermolecular Force Constants 69 C, Space Group Symmetry Analysis 71 D. Cartesia.n Symmetry Coordinates 77 E. Pha.se Dependent Dynamical Matrices 84 F. Dispersion Curves and Polarization Vectors 87 G; One-Phonon Frequency Functions 92 H. Two-Phonon Frequency Functions 93 I, Directional Debye.Wa'ler Factors and Weighted Frequency Functions 93 11i

TABLE OF CONTENTS (Concluded) Page 6. POLYVINYLCHLORIDE 108 A. Molecular Structure and Coordinates 108 B. Intramolecular Force Constants 113 C. Line Group Symmetry Analysis 115 D. Cartesian Symmetry Coordinates 118 E. Phase Dependent Dynamical Matrices 120 F. Dispersion Curves and Polarization Vectors 123 G. One-Phonon Frequency Functions 123 H. Directional Debye-Waller Coefficients 134 7. POLYTETRAFLUOROETHYLENE 135 A. Molecular Structure and Coordinates 135 B. Intramolecular Force Constants 136 C. Dispersion Curves 138 8. CONCLUSIONS AND DISCUSSION 140 A. Polyethylene 140 B. Polyvinylchloride 147 C. Polytetrafluoroethylene 148 REFERENCES 149 APPENDIX 152 iv

LIST OF TABLES Table Page I. Polyethylene Repeat Unit Dimensions 38 II. Polyethylene Chemical Repeat Unit Internal Coordinates 38 III. Vh Line Group Symmetry Elements 43 IV. Star Group Multiplication Table for Single-Chain Polyethylene 44 Vo Star Group Character Table for Single-Chain Polyethylene 45 VI. Reducible Representation Characters for Single-Chain Polyethylene 46 VII. Star Space Group Elements for Crystalline Polyethylene 73 VIII. Star Space Group Multiplication Table for Crystalline Polyethylene 74 IX. Star Group Character Table for Crystalline Polyethylene 75 X. Polyvinylchloride Repeat Unit Dimensions 109 XI. Polyvinylchloride Chemical Repeat Unit Internal Coordinates 110 XII. Polyvinylchloride Line Group Symmetry Elements 116 XIII. C2v Star Line Group Multiplication Table 117 XIVo C2v Star Line Group Character Table 11.7 XV. Polyvinylchloride Directional Debye-Waller Coefficients 134 XVI. Planar Teflon Repeat Unit Dimensions 136 v

LIST OF FIGURES Figure Page 2.1. Momentum transfer vector orientation angles. 12 2,2, Equilibrium position of a nucleus in the ~th unit cell. 20 4.1o Polyethylene chain structure. 37 4v2, Repeat unit cartesian coordinate systems for polyethylene chain. 40 4.3. Single-chain polyethylene symmetry operations. 44 4.4. Dispersion curve branches for symmetric and anti-symmetric species. 54 4.5. Single-chain polyethylene dispersion curves. 62 4.6. Deuterated single-chain polyethylene dispersion curves. 63 4.7~ Single-chain polyethylene longitudinal one-phonon G-function, GI(E). 64 4 8. Single-chain polyethylene transverse one-phonon Gfunction, GI(). 65 5.o Polyethylene crystal structure and intermolecular "Internal" coordinates. 67 5.2~ Intermolecular forces on hydrogen atoms in a CH2 unit of crystalline polyethylene. 69 5.35 Star group symmetry operations for crystalline polyethylene, 73 5.4. Crystalline polyethylene low frequency dispersion curves (FT =.05 mdyne-A/rad2). 88 5.5. Deuterated crystalline polyethylene low frequency dispersion curves (FT =.05 mdyne-A/rad2). 89 5.6. Squares of the polarization vector magnitudes for v5 and v9 hydrogen motions in normal crystalline polyethylene vs. phase angle (6c). 90 vi

LIST OF FIGURES (Continued) Figure Page 5.7. Squares of the polarization vector magnitudes for v5 and v9 carbon motions in normal crystalline polyethylene vs. phase angle (5c). 91 5.8. Transverse one-phonon hydrogen G-function, 4f(C), for normal crystalline polyethylene. 94 5.9. Longitudinal one-phonon hydrogen G-function, GL(E), for normal crystalline polyethylene. 95 5.10. Transverse one-phonon deuterium G-function, G (E), for deuterated crystalline polyethylene. 96 5.11. Longitudinal one-phonon deuterium G-function, GL(e), for deuterated crystalline polyethylene. 97 5.12. Transverse two-phonon G-function, GI(c), for normal crystalline polyethylene as a function of temperature. 98 II 5.13. Longitudinal two-phonon G-function, GL (I), for normal crystalline polyethylene as a function of temperature. 99 5.14. Calculated directional Debye-Waller coefficients as a function of temperature, for crystalline polyethylene. 101 5.15. Calculated directional Debye-Waller factors as a function of temperature, for crystalline polyethylene (Ei = Ef =.030 eV; Gs = t/2). 102 5.16. Calculated longitudinal to transverse elastic peak ratios as a function of temperature, for crystalline polyethylene (Ei = Ef =.030 eV; Gs = </2). 103 5.17. Sum of transverse one- and two-phonon frequency functions for crystalline polyethylene weighted by the Debye-Waller factor (Ei = Ef =.030 eV; Gs = </2, T = 93~0K). 104 5.18. Sum of longitudinal one- and two-phonon frequency functions for crystalline polyethylene weighted by the Debye-Waller factor (Ei = Ef =.030 eV; 9s = t/2, T = 93~0K). 105 5.19. Sum of transverse one- and two-phonon frequency functions for deuterated crystalline polyethylene weighted by the DebyeWaller factor (Ef =.030 eV; 0s = </2, T = 93~K). 106 vii

LIST OF FIGURES (Continued) Figure Page 5.20. Sum of one- and two-phonon frequency functions for unoriented deuterated crystalline polyethylene weighted by the Debye-Waller factor (Ef =.030 eV; Gs = t/2, T = 93~K). 107 6.1. Syndiotactic polyvinylchloride chain structure and internal coordinates. 108 6.2. Polyvinylchloride repeat unit coordinate systems. 112 6.3. Single-chain polyvinylchloride symmetry operations. 114 6.4. Single-chain polyvinylchloride dispersion relations (FT =.06 mdyne-A/rad2). 124 6.5. Single-chain polyvinylchloride transverse (HI or H2) hydrogen frequency function. 126 6.6. Single-chain polyvinylchloride longitudinal (Hi or H2) hydrogen frequency function. 127 6.7. Polyvinylchloride hydrogen (Hi or H2) frequency function for unoriented single-chains. 128 6.8. Single-chain polyvinylchloride transverse (H3) hydrogen frequency function. 129 6.9. Single-chain polyvinylchloride longitudinal (H3) hydrogen frequency function. 130 6.10. Polyvinylchloride carbon frequency function for unoriented single-chains. 131 6. 11. Polyvinylchloride chlorine frequency function for unoriented single-chains. 132 6.12. Polyvinylchloride total frequency distribution, G(e), for unoriented single-chains. 133 7.1. Single-chain Teflon dispersion curves for planar approximation. 139 Vlll

LIST OF FIGURES (Concluded) Figure Page 8.1. Frequency function, G(c), for unoriented normal polyethylene (experimental points from Ref. 5 for Ef =.030 eV, T = 1000K, Gs = 90~; G(c) = 1/3 GL(G)+2/3 GT(c)). 144 8.2. Longitudinal frequency function GL(e), for oriented normal polyethylene (experimental points from Ref. 5; Ef =.030 eV, T = 100~K, Gs = 900 ). 145 8.3. Transverse frequency function, GT(E), for oriented normal polyethylene (experimental points from Ref. 5; Ef =.030 eV, T = 100~K, Gs = 90~). 146 ix

ABSTRACT The incoherent and coherent differential neutron scattering crosssections for crystalline polymers are written in terms of the amplitudeweighted directional frequency functions. These functions describe the dependence of the cross-sections on the vibrational frequencies and displacements of the scattering nuclei. For the sth scatterer, the onephonon directional G-function is: GOS ( e ) = N )(c C (q)) jq jwhere there are N values of q, the phonon wave vector, for a crystal containing N unit cells. In this analysis, only phonon wave vectors directed parallel to the molecular chain axes are considered..yS(q) is the polarization vector of the sth nucleus in the Pth direction, for the phonon in the jth branch of the dispersion curves with wave vector q. The summation is performed over those values of q and j for which the neutron energy transfer is C =:/ o j(q), where cj(q-) is the frequency of the phonon. The two-phonon G-functions are also defined and the orientation averages are performed, for both unoriented and stretch-oriented long chain polycrystalline polymers. The orientation average of the Debye-Waller factor is performed separately from that of the remainder of the cross-section. The validity of this rather common approximation is evaluated. The vibrational frequencies and polarization vectors, which are required for the construction of the directional frequency functions, are obtained from the solutions of the dynamical matrix equations, which are written in terms of cartesian space group symmetry coordinates. To accomplish this, coordinate transformations are performed on published intramolecular and intermolecular internal coordinate force constant matrices. In addition to a detailed discussion of the symmetry analysis and force constan-t transformation for each of the three polymers polyethylene, polyvinylchloride and polyt+etrafluoroethylene, the following results are displayed: I. Single-Chain Polyethylene a. Normal and Deuterated Polyethylene Dispersion Curves b. Normal Polyethylene Transverse and Longitudinal OnePhonon Frequency Functions x

II. Crystalline Polyethylene Lattice a. Low frequency Normal and Deuterated Polyethylene Dispersion Curves bo Polarization vector squares for Normal Polyethylene torsion and stretch-bend dispersion curve branches as a function of chemical repeat unit phase angle, in graphical form c. Deuterated Polyethylene Polarization vectors as a function of chemical repeat unit phase angle in increments of t/5, in tabular form d. One-phonon Transverse and Longitudinal Frequency Functions for both Normal and Deuterated polyethylene over the entire frequency range e. Low frequency two-phonon transverse and longitudinal Frequency Functions f. Directional Debye-Waller Coefficients as a function of temperature III. Single-Chain Syndiotactic Polyvinylchloride a. Dispersion Curves b. Polarization vector squares as a function of the chemical repeat unit phase angle in increments of -t/5, in tabular form c. One-phonon transverse and longitudinal Frequency Functions for hydrogen motions d. One-phonon unoriented Frequency Functions for carbon and chlorine motions eo Directional Debye-Waller Coefficients at, liquid nitrogen temperature IV. Dispersion Curves for Single-Chain Polytetrafluoroethylene (Planar Approximation) In addition, the normal polyethylene composite one- and two-phonon frequency functions, which are weighted by the Debye-Waller factors, are compared to published frequency spectra obtained from experimental crosssections for a stretch-oriented sample. The comparison of the calculated polyvinylchloride frequency functions with experimental values must however await the results of experiments which are now in progress. xi

CHAPTER 1 INTRODUCTION Microscopic neutron scattering cross-sections, for neutron energies much higher than the chemical binding energy of atoms in a. molecule, can be ca.lcula.ted assuming that the atoms a.re essentially free. On the other hand, when the neutron energy is of the order of the energy of vibration of the scattering nuclei, it is necessary to carefully consider the molecular dynamics in cal.culating the cross-sections. This must be done, for example, in calculating the low energy neutron scattering cross-sections to be used in studies of neutron therma.liza, tion in nuclear reactor materials. From a more broad viewpoint, knowledge of the microscopic properties of materials, whether destined for use in a, rea.ctor or not, is becoming increasingly important. Such knowledge ca.n be obtained from inelastic scattering mea.surements using low energy neutrons with wave lengths comparable to interatomic spa.cings and energies comparable to vibrational energies. The neutrons then serve a.s probes for investigating the microscopic dynamical properties of the material.l In these experiments measurements of the gain or loss in neutron energies and neutron momenta,, which are compactly described in terms of the differential neutron sca.ttering cross-section, lead to increased knowledge of the atomic vibrational frequencies and displacements. Theoretical ca.lculations of terms related to the scattering cross-section, such a.s those undertaken in this study, should greatly aid in obtaining a. better understanding of the experimental neutron scattering results..1

This analytical investigation was primarily motivated by a. desire to better understand the neutron scattering measurements which have been performed on normal and deutera~ted polyethylene. 1.12 In addition, it wa.s motivated by other scattering experiments on crystalline polyvinylchloride and polytetra,fluoroethy i ene (Teflon) which are either in progress or planned for the near futureo Therefore, the ma.in objective of this investigation was the calculation of those termS in the neutron scattering cross-sections of crystalline polyethylene, polyvinylchloride a~nd polytetra-fluoroethylene which contain informa,tion regarding their atomic motions. To be more explicit this main objective, which was threefold, was to calculate: i. The onea.nd two-phonon directional frequency functions and the DebyeWa.l.'er factors for both norma.l and deuterated polyethylene. 2. The one.-phonon directional frequency functions and Debye-Waller fa.ctors for polyviny'oh>.orideo o The frequency vs. phonon wave vector dispersion curves for polytetrafluoroethylene o The a~dditional, goal wa.s the comparison of these quantities with the a~vaila,ble expecimental resu'Lts, with the expectation thamt this would lead to an increa.sed understanding of the molecular dynamics of these olymers and perhaps eventua..lly to a better description of the neutron scattering interaction itself.

CHAPTER 2 NEUTRON SCATTERING THEORY Before concentrating attention on any one of these molecular crysta.ls, it is first necessary to examine the neutron scattering cross-section and those terms mentioned above which a-re primarily sensitive to the at.omic motionso Since incoherent scattering from the hydrogen atoms in these polymers is predominant, unless deuterium is substituted for the light hydrogen atoms the incoherent cross-section will be discussed firsto Ao INCOHERENT SCATTERING CROSS-SECTION Summerfield has shown'13 that the incoherent neutron scattering cross-section for an atom in a, long chain polymer such as polyethylene is, under certain approximations, proportional to an amplitude weighted directional frequency function, G((E). This G-function for the sth atom, can be written as. * jq where there are N values of q, the phonon wave vector, which is assumed in these calculations to be directed parallel to the polymer chain axes. N is the number of unit cells along this chain direction. y7S(q) is the atom's polarization vector in the Pth direction for the jth branch of the dispersion curves, for wave vector qo The summation in equation (2.1) is over those values of q and j for which the energy transfer, =- j(q), where (j(q) is the frequency of the phonon in the jth branch with wave vector q. 5

4 The polarization vectors, which are mass-weighted displacement vectors, are orthonormal, so C7sO(q)7.s (q) = A (2.2) and ySa (ZS( q),6. (2.3) where the summation on s is over all atoms in a unit cell. The mass weighting takes the form 5( q) = ml/24s(q) (2.4) where jS(q) is the displacement along the ath cartesian coordinate axis for the sth atom. Actual QalculationS are per:formed using the phase shift between adj:acent chemical repeat units, 6-C rather than q or the phase shift between unit cells which is 0 = qc where c is the unit cell dimension along the chain axis direction. Special care must then be taken to see that the eigenvector solutions to the equations of motion are properly normalized* as required by Eq. (2.3). *See, for example, the section on Dispersion Curves and Polarization Vectors in Chapter 5.

5 Since the incoherent scattering is primarily from the protons, the s index is omitted in the following equations, which then refer only to scattering from hydrogen. The G-function is of major importance insofar as it contains both the frequency and displacement information for the crystal. The relationship between this function and the differential incoherent scattering cross-section for a. hydrogen atom is given by5'1213: 00 d2 N kf ao < eZ(o) dt e-ict// eZ(t)> (2.5) dQde ki 4n 2rT -00 since Z(t) is defined as z(t) = 2m Ku d Goa' Jd' c+() X) t +.e E: J 2m El -Pt' ePE'-l (2.6) The symbol < > indicates the usual thermal average and, in addition, an appropriate average over molecular orientations. The initial and final neutron momenta are, respectively, 11ki and Ikf. or is the bound hydrogen atom scattering cross-section. KCa and ca, are the UCth and' th components of the neutron momentum transfer vector K = ki-kf and m is the mass of the hydrogen atom. The neutron energy loss is denoted by E. In the case when A, which is equal to l/kT, is large neutron upscatter in energy is relatively unimportant. That is, a, reduced number of phonons is available for low temperature annihilation events. Then the last term in Z(t) is insignificant and phonon creation events are dominant. It is convenient for comparisons between theoretical and experimental results in hydrogenous materials to introduce a, function:

6 8rn ki c(k -e-( e) d2ainelastic _ --....kf N dd(2 d2a which can be extracted from the experimental results for dad in a neutron downscattering experiment. Stretching a polymer film tends to orient the molecular chain axes along the stretch direction. In an experiment on such a stretched target, the neutron momentum transfer vector K can be aligned either parallel or perpendicular to the stretch direction. For the parallel alignment, the so-called longitudinal G-function is obtained, where5 G,, () GL(e) (2.8) For the perpendicular alignment, the transverse experimental G-function is obtained, where GK (E) -G(E). (2.9) -I In an experiment on. a polycrystalline target, in which all relative alignments between K and the chain axes are possible, the unoriented experimental G-function is measured, where G,(E) - G(e). (2.10) These experimental results are in fact Debye-Waller factor weighted spectra. This can be seen by writing Ga(e) in terms of Z(t) as 00 Gs_(6) = 2 e(l-~6) 2 e Z-(o) dtf e -it//[eZ(t) ]> (2.11) ~~~~~~~~~~~~-1 (22l2

7 and noting that in the longitudinal case, the Debye-Waller factor is e-2WL = < e-Z(~) >L (2o2) while in the transverse case it is approximated by e- 2 Wr;: < e Z(~) (2,13) where 00 2WL = J I Ms cctgh(fZ2) GI(E) de (2.14) 2WL =, 2m ~ 0 and 00 2WT K= cAt2 ctgh(PE 2) GI(e) dE. (2.15) G(EC) and GI(E) are determined from the respective longitudinal and L T transverse orientation averages of the amplitude weighted directional frequency functions described in Eq. (2.1). These averages and the approximations involved are discussed in a. following section. Bo ONE- AND TWO-PHONON DIRECTIONAL FREQUENCY FUNCTIONS The experimentally determined G,(c) function, in general, contains contributions from multiphonon events in which two or more phonons are created and/or annihilated. This is in addition to the more important contributions from events in which a. single phonon is either created or annihilated. In order to evaluate the extent to which two phonon contributions may be included *;See the section on Orientation Averages, this chapter.

in experimental results, the crossesection is written in terms of Sj-olander' s ].6 5 phonon expansion.~ 6 Equation (2.5) then becomes, 00 dS = ~kP a (tZ(o) dt ) Cdr _ N 0 -z<'( 0 2dt - — t/ [1 + Z(t) + Z ] > d~d- k 4 2: 2 (2.16) Z(t) is given in Eq. (2.6) a,s a, function of $ (s' ). This may also be re-wr itten as Z(t) = de' V) f et// (2.17),_/ e where G z(c' ) is the matrix element for the Uth row and Pth column of the matrix G(C'). This latter form is more convenient for performing the orientation averages where,only the terms in [K o.(c' ) o ] are directionally dependent. From Eq. (2.1.6) the cross-,section for an atom, including the two phonon te rm.i.s d2a r d2-a(o) d2((1) d2o(2) (2.18) d-2d-c' d~des + ~_- + _____ dQd c d~1dc C d dQd dQd Since the phonon number in a low temperature target is small phonon annihil.ation produces a. negl.igible contribution to the cross-section. Using Eqg, (2.17)) and performing the time integrations, the contributions for neutron downsrcattering events only are therefore dt~ ) = N~o kf 6(c) < eZ(o) > dQdo 47r ki and

kf/ e ('(e) > (2o20) dd de 2tn 2 d(2) a, kf 2N ) del [w. G(c') ~ m)(l K ~(E~' ) * G E) ] dQdE 4g ki 92mJ 2e (e c' )(1-De-E' )(l-e-P(e-C' )) (2.21) The orientation averages may now be obtained0 Myers has already given the result for the one phonon term, for both a, polycrystalline sample and a stretch~ oriented sample 12 To obtain the Debye~Waller factors, e=Z(~) is separately averaged and is replaced by e < Z() >0 For a. polycrystalline target, the results are 5 < - G(c) o > (222)....p 3 O=l1 and 00 <Z(o)> = 2W de!2 K2 ctgh(Pe 2) GI(e). (2025) P 2m E o For a, longitudinally oriented stretched target there are no approximations in obtaining the results < KD o G( ) > = 2G35() = GI( ) (2024) and WL 2m00 where CO = 5 denotes the stretch direction0

10 For K aligned perpendicular to the stretch direction, the transverse orientation average < e-Z(~) s. G(E). _ > is approximated by separately averaging e-Z(o) to obtain < G() * >Tr = [G11( ) + G22 ( E)] = G(E) (2.26) Tr. 2 T and: < = 2W > W d a ctgh(:/2) G (E) + Go ( (2.27) In the section on Orientation Averages, which follows, this approximation is shown to be valid when Gll(c) - G22(c). The one phonon directional G-functions are therefore defined a,s GI(E) = G33(c) (2.28) GI(E) = 1/2[Gl (c) + G2(E) (2.29) and the polycrystalline one phonon G-function is defined a.s GI(c) = 1/3 GI(c) + 2/3 O~(E) = /3 [Gll(E:) + G22(E) + G33(E) ] (2.30) The Debye-Wa.ller factors have been separately averaged, which is equivalent to replacing the one phonon contribution to the cross-section Z(). ). > (2.31) with

11 e < Z(o)G(). >. (2.32) The accuracy of this approximation will now be evaluated and the two-phonon orientation average will also be obtained. C. ORIENTATION AVERAGES Preliminary calculations of the directional Debye-Waller coefficients, using the approximation that < e -Z(o). G ). > e G( indicate that for 90~ scattering experiments on polyethylene to a fixed final energy Ef = 30 millivolts, 2W is approximately given by 2W O.o01 = 0.01 [2Ef + c] (2.33) 2m so that the range of values of 2W is.6 < 2W < 1.3 for 0 < E < 70 mev. (2.34) The values of 2W for polyvinylchloride, which are given in Chapter 6, are also found to be in approximate agreement with this polyethylene value. Therefore, expanding around 2W = 1, let

12 e -2W eX-1 ex = e(1-2W), 1 + x + x /2 +... for sma.ll x. Then ex 1 + (1-2W) + (1-W)2 =2 2(2W) +(2W) 2 2 2 or e-2W - 2 (2W)+ (2W)2 (2W) + (2W6) e 2 2 This is an excellent approximation with less than 1% error for 2W in the range of 0.6 to 1.3. Therefore the average can be written as <-e-Z(o) 1 Z2 (o). _(o). G].- _(e) > < [ - 2Z(o) + G(E) ]> e' 2 (2.37) where. G(e). _ = -2Gll cos2 4 sin2Q + G22 sin2 B sin2g + G33 cos2@ + 2G12 cos > sin > sin2 + 2 G13 cos cos 0 sin Q + 2 G23 sin sin G cos @3 (2.38) with 0 and, as shown in Fig. 2.1. -3 2 Fig. 2.1. Momentum transfer vector orientation angles. Z(o) 2 m - - - 0~

.13 In the longitudinal case, no orientation a.verage is required. However, using these relations, (2.38) and (2.39), the transverse orientation average, for 9 = T/2, may be written -Z(O).2. >2 (e - 2e h ctgh(E:'/2) -- e >6 2 e) < Tr. 2e'm E x [Gll(' )cos2 ~ + G22(E ) sin2] x [Gll(E)cos2~ + G22(E) sin2~] > + _ 3) 2 < mdEctth(/'2) 2e 2m x [Gl1(C' )cos2~ + G22( E )sin2M] 00 /do"<, ctgh(c" /2) GA11(El )cos26 + G22(c" )sin2][Gll( E)cos2 + G22( 2)sin2] > 0 (2 40) where G12(E) is calculated to be zero for all values of E. Since the aligned crystallites ha.ve random orienta.tions about the stretch direction axis, the integration over the angle ~ is performed and the result is: 00 < eZ(o). G(f) s >Tr G -e 2G(e) - m 4 d e Tr. 2e T e2m 0 X (Gl (')[2G"11(E) +.22(E)] + G22(e j [G11(e) + 3J22(E)]J 00 00 + 2 ) dE' de ctgh(:c'/2) ctgh(e /2)(Gll() Gll(" ) [-G(E) o o + 1 G22,E)] + G11(E' )G22(c ) [G11(') + 22(E)] + G22(et )Gll(I) [Gll(E) + 122(E) -+ G2l2(E&)G22(et),, -ll(e) + 22 ). (~ Perorming the energy 16ntegra.ons leas to Performing the energy in~tegrations lea~ds to

14 < e-Z(o) __ Tr.~ 2e ~2G2(c) ) _ 22 + (2W22(e) >(2w)[L() + (w <c- e ( ) _ G(E) 0 K 25 - 2 + (2W22)[ l( ) + 8G22( ) ]1 + 2 ((2W11)2L5ZG"(e) + G22(e)] 8 8 2e 16 lb + (2w ))(2 [Gll(E) + G22(c)] + (2W22[) I Gl() + G22(E)]+ (2.41) ) 8 iLL 22 22 E6 16E Assuming that 2Wll - 2W22J which is a good approximation for polyethylene, the average becomes e =Z(o)o G(2) 2 > Tr, 2-2 GT( ) e2 (2Wll)[ ) 2) + 2 w G)(c) + G22(c) ) 2 ((2W1 )2). 2e -' 2 or~ < -()K G(c) OK > ~ (WTr., eel2e > +e Ge(E). (2W42).... 2e T But this is < e.()'G(E) o K >Tr <:::Z(O) ><._ — Tr.(243) providing that the value of (2W11) can be replaced by (2WT), which follows if G22(c) G G11(c) in: GI(E) Gll )G1 (2-44) T 2 and 2W =i/22 de' S ().. (2.45) T 2rm eT 0~

15 The requirements and approximations involved then in replacing <eZ(o)Z G(z) (o > by < Z(o) > <' G() > for the range of E and K2 of interest here, are that G12(c) = 0, for all C 2W11 2W2. Gl l(c) G22(c), for all c (2o46) The first requirement is completely satisfied for polyethylene and for the second, the discrepancy between 2Wll and 2W is less than 2% of the larger value. 22 The third requirement is not satisfied for all C since the difference between G11(c) and G22(c) is as much a.s 8% of the larger value at some Co The two phonon orientation average will now be obtained, assuming again that the Debye-Wa.ller factor may be separately a.veragedo Then < [7s_ ~(c) o r_ ]2> = < [ A2(Gll cos2a sin2G + G22sin2.sln2ao+ + G33 cos20 + 2 G12 cos ~ sin ~ sin2o + 2 G3co os sin 0 + 2 G23 sin ~ cos g sin 0 ]2 > (2~47)

16 In the longitudinal case, the result is < [ ) ]2 > _ [ 2eGI(~I ^GI(c-) J (2-48) - L L L For the transverse orientation, when g = i/2, the average is 2rt 1 _ d~ [ I2(Gll(e' ) cos2~ + G22(E') sin2~ + 2 G12(C') cos ~ sin ) ]2 2T[ 0 (2 49) but G 12() is zero for all eo The result is [ _~G(~ ) Gll P:, 22, <Kr o [ K) 3_0( > )= ( ( G11(I" )l + Gl(c) ( (E) + - G (E')Gll( e) + G22(3 )G 22(E11)] (25o0) 8 8 Since GI(')EGll(e )+ G22(E) (2.51) T 2 then T T 4 G4 ( )I(~" l ) + G22(E' )G22( s") + G11(E' )G22( c') + G22(E' )Gll(c")}. (2.52) Notice that this is not identical to the two phonon orientation result in equation (2.50), so that the statement < [ G(' ) ][. (E ). -] >Tr. 4GI()GI() (2o53)

17 is valid only if Gl1(e), G22(c), for all C. This is the same requirement as that for the validity of the separate averaging of the Debye-Waller factor. For an unoriented polycrystalline sample, the two-phondn orientation average must be performed over both ~ and g angles. Again, if G12(c). - 0; the polycrystalline average of <[ o _G(' ). K ] G(oc") E K] > (2o54) is < K 4[Gll( )Gll(c")cos4 sin4~ + G11(' )G22(c") oso2 sin24 sin4g + G"(E' )G33(c" ) cos2~ sin2Q cos2G + 2 G1(E' )G1O3(c" ) cos3 Cos 0 sin30 + 2 G(llE )G23(e ) cos2* sin > cos g sin3g + G22(E'&)G11(c") cos2 sin22 sin40 + G22() ( )2(c) sin4 sin4g + G22(e t)G33(c" ) rin2? sin2G icos + 2 G 2(E' )GI3( _" ) sin2~ cos cos s sin3G + 2 G22( )G2(c") sin3 cos 0 sin3G + G33(E' )Gll( "t) cos2 sin2Q cos2G + G33(c, )G22(T,,) sin2~ sin20 cos20 + G33(' )G3(") cos4 + 2 G33(' )G1 ( ) cos coso3 sin G + 2 Gz53(e )G23( ll) sin / cos30 sin 0 + 2 G3(t,' )Gul1(,,) cOs3 sin3Q cos 0 + 2 G13(c' )Go22(t, ) sin24 sin3g cos 0 + 2 G13(e' )G33(c" ) cos50 cos 4 sin 0 + 4 GL3(C' )G23(C") cos ~ sin ~ cos20 sin2Q

.].8 + 2 G23(' )Gll( cos2 sin sin-3 cos 0 + 2 G23(c' )G22(E) sin3 cos sin30 + 2 G253(-')K33(e") sin b cos3Q sin G + 4 G23(c' )GL3(c") cos sin ~ cos2g sin2Q] > (2o55) The final result, after integrating over 0 and ~, is < [ G(E' ) [ E" ). i > - [ (G( )Gll( ) + )G. 1 G22( G )Gll1(h t,) +. G22(E, )2(c) ] + [G33( )G33(c ) ]' 8 8 +..I [G11( r )G:5( cT + G22( c )G(et) + t( )G31(cf,) + G &'(c, )G22( (") 1] 1.6 (2.56) Then the two-phonon unoriented G-function is given by ~i~II~c) 3 II(I I I 7( " GT ~(CG ) +'GIL (e) (2. 57) 8 T (E) + TTL where if -" - (c -' ), Eqso (2.20), (2.21), and (2.56) may be used to define the two phonon directional G-functions as _______( I __) _ C _ d ___ [ v1 G "+ l'G22-(c' T~f 2, \2i/ 8 ~ l~~1lT2 Gii(1') -.(l"e=) 2,, )1 G22(, )] 8 8 GT (2C )([e..( )) 2(5(l ) ))( LG ). 8(, ) 2 1

19 00 GII () e:(l-e-E) (.2 kc? TL 8 2m 0 o de Gll(E )G33( C" )+G22(E' )G33(e )+G33( e )Gll'() +G33(, )G22(, F' (-c' )(l-e-P(E-', If GG (E) I G22(e) for all e, then these equations may be rewritten a,s II 2 (l') (~c' II(e) - e(1-eleP)( 2) d ) G(e-e' ) 2 2m ) E (-e' l-eE )(leeE(E)(-e Ec )) GIL (~) = ) (?'et 2m 1,'t ) II(E) E(eE) 2 2 Gj(o )G ( L 2 2m E' (E-') )(1le-E3 )(-e' )) 0 -e ~L 4 ~~2m o (-e' )( -(~-' )) G$I(c ) - c(1-e-t~C).<2,tG_~ e )GL~e-E.... - (2.59) These equations are used in evaluating the two-phonon G-functions in this report. The unoriented two-phonon G-function is approximated by GII( E) 1 GII( E ) + -I() (2.60) 3 3 rather than by Eq. (2.57), since this is more convenient due to its similarity with the one-phonon unoriented G-function. This step is partially justified by experimental results12 in which the sum 1/3 G(E) + 2/3 G(E) from para,llel and transverse oriented experiments matches the unoriented results for the energy region in which two-phonon effects are believed to predominate. D. INCOHERENT APPROXIMATION FOR COHERENT SCATTERING CROSS-SECTION The coherent neutron scattering cross-section may be written as1516 d2 o-ld- k 7 a isas~ (r~s -r~s ) e-1/2 [Z~sS(O) + Z,,(o) d~dE k1L~ a~i,a 00 - - 2~ e 1s' dt (2.61) QC _0O~1 2~H

20 where the equilibrium position of the a'thnucleus in the Ath unit cell is RI + rS' as shown in Fig. 2.2. RI is the position of the Ith cell relative to the crystal coordinate system origin at the zeroeth unit cell and ros is the position of the s' thatom relative to the origin of the ~th unit cell. The coherent scattering length for the sth nucleus is as. 1 Fig. 2.2. Equilibrium position of a. nucleus in the ~th unit cell. In the above equation, Z~s (t) is given by Is' z0s (t) =- d K K de e,t/i + e I?' 2(msm )1/2 a,~1-~P s' S S' -e e, (2.62) This reduces to Z(t) in Eq. (2.6) for the incoherent case, when I = 0 and s = s', since the G-function in Eq. (2.62) is defined as G os iq. Gos0 (IE) = I yosa(q) ys'C' (q) e * R -(e (q)) (2.63) N 7j J for e > 0, where GosSa' (-I ) os=c* (+E,).

21 For K large enough so that the changes in K which accompany changes in c are relatively small, ~ can be treated as a continuous variable and the sum Na -1 ~N 1Nc-' i, R2 +IC3.3 eis >R =), a, + 2, Ib + K I ci (2.64) e 11' 22 5 ~ ~ 1'-0'~ 2=0,-l 3=0 can be replaced by its average over values of Ko Since 03 1_j d.K eil( la) a, 2Tc -00 the average value of each term in the sum Na-1 N a -l j eiKlel e I1 =0 -1-1 e ei1~1a = (2a65) or I e01 A similar result is obtained for e — (r r (2.66) ss Therefore the cross-section Eq. (2.61) may be rewritten as 00 s o00

22 which is of the same form as Eqo (2.5) for the incoherent cross-section. When r is large and ~ = 0 as required by Eq. (2.65), Eq. (2o63) is GocT (E,) 1 ( q) os) c (q) e( oj( q)) (2.68) osO' N= j (- q 7qj which is the same as Eq. (2.1). Therefore when K is "large," the coherent cross-section can be written in terms of the phonon expansion as in Eq. (2.16) for the incoherent cross-section. The directional G-functions in the incoherent approximation for the coherent cross-section cain also be defined as they are in Eqs. (2.24) and (2.26).

CHAPTER 3 LATTICE DYNAMICS The vibrational frequencies and ca.rtesian displacement vectors necessary for the construction of the directional frequency functions ma~y be obtained from the solutions to the equa.tions of motion for the system. The usual method would be to first solve Wilson' s GF matrix equations718 for the displacements in internal coordinates followed by a transformation to cartesian coordinates. - However, due to the large number of cartesian eigenvector solutions required in this a.nalysis, a, cartesian coordinate dynamical matrix equa,tion is first obtained so that only one transformation between coordinate systems is necessary. Ao DYNAMICAL MATRIX EQUATION In matrix notation, the potential energy of vibration, V, is given by~ 2V RtFR (3 1) where F is a matrix of internal coordinate force constants with matrix elements, in the harmonic approximation for the potential, given by' f1i aR 6 R 23

24 the' matrix and R vector are of order 3mN x 3mN and 3mN, respectively~* The internal coordinate displacement vector and a cartesian displacement vector, X, are related through the equation: R = B X (35) -where the elements of B are determined by the geometry of the crystal. In internal coordinates, the kinetic energy is given by: 2T t R G R (3~ 4) whe:re G B ML' Bt (3 5) and M -1 is a diagonal. ma.trix with elements equal to the reciprocal ma.sses of all atoms in the system~ Making use of Eqs. (3.1) and (353), the potential energy in cartesian coordinates is given by'19 2V Xt t F B X (X>~6) which may be:rewritten a.s: 2V xt F X (3;7) where the cartesian coordinate force constant martrix is: F Bt tFB (3i8) ~:Neg.l.ec-ting redundant coordinates

In cartesian coordinates, the kinetic energy is given by: 2T = XtMX (3 9) where in analogy with the definition of G in Eqo (3.) the G matrix in cartesian coordinates is defined as: G - M-1 (j3 10) Notice that this is a, diagonal matrixo There is a system of normal coordinates in which F is also a. diagonal ma,trix. Therefore: X Lc (3.11) where Q is the normal coordinate vector and L is the transformation matrix. Then: 2V = Qt t F. L Q (3 12) or: 2V Qt A Q (3.13) where: Ae kin Lt'F. iti ( 3.4) The kinetic energy may also be written using Eqs. (359), (3>10), and

26 (3 11) as:~ 2T t t G,-' L Q (3 15) where: Lt G1L = E (3e.16) Since E is the unit matrix: Lt L1 1G (3517) and so fina.lly Eq. (35.14) becomes: L-1 G F L = A (3.18) or: G Fc Lc L A (3.19) which is analogous to the Wilson GE equation in internal coordinates. The matrix A is a, dia.gonal matrix whose elements are the squares of the frequencies of vibration and L is a matrix of column vectors whose components are the C atomic displacements in cartesian coordinates. To facilitate solutions for A a.nd L, a symmetrical form for G F. is obtained.20 Then. ---- ~-C =C=c CT = r A (3.20) where

27 C = /2 Fc Gl/2 (32L1) re -G/2 L (3~22) -c cWC -c and G1/2 G1/2 G (j 323) -c.Cc 3 The ~ matrix contains column vectors with components equal to the cartesian =c atomic displacements, each weighted by the square root of the atomic massO These are the polarization vectors for which r t = E, in agreement with Eqo (3.16)o It is the dynamical matrix Eq.,( 320) which must be solved for the eigenfrequencies and eigenvectorso Bo CARTESIAN SYMMETRY COORDINATE TRANSFORMATION The use of the space group symmetry of a crystal makes it possible to perform a transformation to cartesian symmetry coordinates which block diagonalizes the dynamical matrix, C.o All symmetry elements in that subgroup of the space group which leave the phonon wave vector, q -", qc invariant may be 21 utilized (In this report, the wave vector is assumed in each case to be directed along the molecular chain axes which are parallel to the unit cell c axis ) The symmetry coordinate displacement vector is S, where' S = U (X324)

28 and U is the matrix of coefficients for the transformation. Then the dynamical matrix equation becomes: or: Cs P5= IC A (3.25) in cartesian symmetry coordinates, where: cs = u c u. (3 o26) For convenience in the computer calculations, U is written as a product of a phase dependent matrix and a, diagonal matrix, I(+), whose elements are appropriate powers of (-1) required for the symmetry coordinates. The phase re-ferred to is that between adjacent chemical repeat units. It is: e - io2 when c is the unit cell dimension along the chain axis direction and there are two chemical repeat units per unit cell, which are relate-d by a. glide or screw fractional translation. Therefore: U - U (c) U (+) (3.27) and rs = u(c) u(+) r (3.28) ~C ~ -_C

29 The block diagona.lized dynamical ma.trix may then be written as~ cs = U (bC) CQ(+) U(c)t (3~29) where: c(+) = u(+) c u (+) (3 0):= iii — (3o-30) The last transformation is performed by using~ Bs B U(+ _ (3o31) so that: ) = (+) G1/2F G1/2U(+)t (3 32) or: c(+) G/2[BstFBs]] G./ (3 33) The above treatment is adequate when performing a, single-chain analysis using only intramolecular force constants and one-dimensional space group (line group) symmetry~ In the general case, when intermolecular forces are also included: F = BtFB + B t F'B' (3 34) where the intermolecular transformation matrix B' is defined by: R' = B'X (3535)

3o0 and F' is the internal coordinate intermolecular force constant matrix. The result is that when intramolecular and intermolecular forces are both used and three dimensional space group symmetry is employed, the dynamical matrix equation to be solved is Eq. (3.25): Cs rFs = Fs A C=fC ='A where~ c = u(_ ) g(+) U(6c as before, but now: _c(_.+) _ (+) Gl-/2 FC../21(+)t (3.36) with F given by Eq. (3.34) so that: c(+) = G1/2 [Bst F Bs + B' St F' B' s G1/2 which is written as: C.(+) Gl/2 [A + A ] Gl/2 (37) where: A Bst F Bs (3 o38) and: A' B' t F' B' s (3,39)

3.1 These a.re, respectively, the cartesian intramolecular and intermolecular force constant matrices. Since U(bc) and G1/2 are both diagonal, CS may be rewritten as:. wc cs = G1C/2 U(6C) [A + Al' ] U(o)t} G1/2 (3.40) or finally a.s: cs Gl/2[A(6c) + A'(oc)] G=I/2 (.41) where: A(bc) - U(bc) A U(6c) (3 42) and: A'(6c) U(6c) A' U(6c)4 (3 45) C. CARTESIAN FORCE CONSTANT MATRICES The intramolecular and intermolecular force constant matrices, F, and F', may be transformed to cartesian coordinates as indicated in Eqso (7338) and (3539)~ Consider, for example, the intramolecular cartesian force constant matrix o A Bt= F B = B5 F B (344) where the matrix elements of Bs are real numbers,

32 The F BS, and BS matrices are partitioned into submatrices, a.s shown in Eqs. (3.45), (3.46), and (3.47). Each F and Bs submatrix contains (3 h + no. redundant coordinates) rows, where h is the number of atoms per chemical repeat unit along a chain axis. The index n labels a chemical repeat unit. Then: n-3 n-2 n-l Rn n+l n+2 n+3 n+4 n-3 ". n-2 F n-2,n n-1 F F =n-l,n _n-l, n+l F=Rn F F F F F =n n,n-2 =n, n-l =n,n rn, n+l la, n+2 n+l N F FF F F n-n+l,n-l Fn+l,n Fn+l,n+l _-n+l, n+2 Fn+l,n+3 n+2 F F F F n+ n+2, n =n+2, n+l n+2, n+2 n+2, n+3 =n+2, n+4 n+3 n+3 |.*... n+4 (3545)

33 n-3 n-2 n-l Xn n+l n+2 n+3 n+4 n-4..... n-3 n-2 Rs =n-2., n n-1l B3 Bs Bs B -n-l,n-2 Bn-l, n-l n-l,n L ln+l Bs R s s B=Rs B B Bs B =n,n-1 =nn fn,in+l nn+2 ~n+2 |o~ ( X \ ~=n+ln n+2 () -n+2,,n+1 n+3 (3.46) n-4 n-3 n-2 n-1 Rn n+l n+2 n+3 n+4 n-3 n-2 n-1 Bs =n+.l, n BS' S BB B = Xn |n BX Bs = Biann+2 =n, n+l nn+g B3 =n-l, n * n+2 \ n+3 (3.47)

34 In Eq. (v.47) the notation Bs +1 denotes the transpose of the submatrix 1- =n, n+l found in the nth row and (n+l)st column of the Bs matrix. From Eq. (3.44), a, submatrix in the partitioned A matrix is given by: An~ X(3 )~ mX: B s (3.48) This is the submatrix for the negative of cartesian forces on atoms in the nth chemical repeat unit resulting from unit positive displacements of atoms in the mth unit. In this equation, (BS) is the submatrix found in th6 nth row and n,O Xth column of BSo From Eq. ( 347), this n, 0th submatrix of the transposed Bs matrix is the transpose of the C, nth submatrix of the Bs matrix. Since: F F F (349) =r, c ='o c-r -n, c-r+n and: F F nn+l _n, n-l Eq. (3o48) becomes: A (Bs) F B S (3 50) -n,m = ( -o x (3,50) Further, since: s = Bs =r, c -o, c-r then if:

35 7 = -a the result, for forces on the zeroeth unit, is: A = 7 ) F B (351) -om g-(- o, -oy -o,Om-7-QZ The limits on the indices a and y are determined by the range of the internal coordinates and nonzero forces assumed in a given model. Usually these limits are determined by the internal rotation (torsion) force constant, since changes in this coordinate affect atoms at the greatest distance from the zeroeth unit. Do PHASE DEPENDENT MATRIX EQUATIONS The A and A' submatrices are obtained from the set of internal force constants through transformations which are described by Eq. (3.51). The phase dependent matrices A(b) and A'(6) are then given by Eqs. (3.42) and 3.43). Using Eq. (3.41), the block diagonalized dynamical matrix C~ is then obtained. There will be N phase dependent blocks, where N is the number of unit cells along the direction parallel to the chain axes. That is, there are N values of q for - c < q< + c or -- < O < i, where the phase between unit cells 0 = qc. Since the space groups used in this report contain twofold screw-rotation or mirror-glide elements, there is a further block diagonalization of each (3m x 3m) phase dependent block into smaller matrices, where m is the number of atoms in a unit cell. These smaller blocks are identified in terms of the phase between adjacent chemical repeat units, 6 = 0/2. c

36 These observations are amplified in the individual sections in which phase dependent dynamical matrices are discussed for the various molecular crystals. See for example the section in Chapter 4, for polyethylene.

CHAPTER 4 SINGLE-CHAIN POLYETHYLENE An extended transplanar model of the polyethylene chain is used, as described below. A. CHAIN STRUCTURE AND COORDINATES A single chain of polyethylene consists of a planar zig-zag configuration of CH2 chemical repeat units as shown in Fig. 4.1.22 +2 +~11 1 I DIHn+2 Cn+2 1 H11 H1 + 1 n+2 2 7 Cn+1 +l -n+ /n+l/2 O H1 ~~nn-n nFig. 4.1. Polyethylene chain structure. 37

38 The interatomic'bond lengths and angles are given in Table I.23 TABLE I POLYETHYLENE REPEAT UNIT DIMENSIONS rCH = 1.o07 rcC - 1o533A c-c-C =;11154t 9H-C-H = 1070 aH-C-C = 109028' C = 2.54A The internal coordinates for each chemical repeat unit24 are described in Table II. TABLE II POLYETHYLENE CHEMICAL REPEAT UNIT INTERNAL COORDINATES Coordinate Atoms Type r n/2 Cn-Cn+1 stretch rH1 Cn-H1 stretch rH2 Cn-H2n stretch -Gnh Cn+l-CnCn-L bend Gn Hn-Cn-Hn 2bend a Cnl-Cn n nbend e2n Cn_1-Cn-H2'bend C3n Cn+l-Cn-Hn bend an Cnll-CnHl n bend T'n+l/2 CnCn- l torsion

39 A (10 x 9) submatrix of the partitioned Bs matrix, which relates the change in the internal coordinates of the nth chemical repeat unit to unit displacements in cartesian coordinates for the (n+l)'st unit, is shown in Eq. (4.1). Ain+i n+il AAnYnl + AZn+l AXn+I AYn+ AZ n+H Arn+l/2 ArH1 n ArH2 n An AGn Bs c= 10 x 9 — n,, n+1l n AU2 Ana ATn+i/2 (4.1) Only the row and column labels are shown. The actual intramolecular B matrix elements, which are obtained in a computer program using a method similar to that employed by Overend and Scherer,25 are shown in the Appendix. For singlechain polyethylene, the chain-oriented cartesian coordinate systems shown in Fig. 4.2 are used. The equations on pages 55-61 of Wilson's bookl8 are used to calculate the B matrix elements, except that a negative sign between (e12 x e23) and

40 (e43 x e32) is changed to a positive sign26 in equation (22). In addition, the torsion coordinate: AT = Ti/9 (4.2) i=l for nontetrahedral geometry is used.26,27 The submatrices,, cB which obey the relation Br = B as mentioned previously, are obtained using Eq. (3.31), which is: B = B U(~. This transformation is effected by simply applying the (~) signs to the matrix elements of B as required by the symmetry coordinates, which are described in a following section. These required sign changes may also be viewed in terms of alternating cartesian coordinate systems which have like orientation in each CH2 unit relative to the H1 and H2 atoms. These right-handed coordinate systems are shown in Fig. 4.2. Hi XH HI H y Fig. 4.2. Repeat unit cartesian coordinate systems for polyethylene chain.

B. INTRAMOLECULAR FORCE CONSTANTS In terms of the internal coordinates, one of the 10 x 10 submatrices of the partitioned 20N x 20N force constant matrix, F. is: Arn+/2 Ar H+ ArH+ Am+j An +1 a~l n+l A+ n+l - 3 2 Arn+l/2 Hi Ar1 ArH2 n Fn+l AGn Aca n (4.3) which again indicates only the row and column labels. The actual matrix elements are taken from the work of Schachtschneider and Snyder,28 with the exception of the torsion force constant for which a value of 0.107 mdyne-A/rad2 24 was substituted. These intramolecular force constant matrix elements are given in the Appendix. Each of the 9 x 9 submatrices of the partitioned cartesian coordinate force constant matrix, A, will have labels as shown in Eq. (4.4).

42 Ac AYc Ac A Hi yHl Hi H2 n2 H2 c n+l nYiZn+i AZnl1 n+ n+ n+ AXn+ AXH AZn AXnH2 (4.4) The submatrix A is given by Eq. (3.51), where the choice of force constants di.ctates that: -2 < m < +2 Huits as far away as (n+2) and torsion coordinate changes in unit (n-2) affect atoms in unit n. Other limits on coordinate changes in the nth repeat unit affect atoms in units 2atoms in unit n< -2 (m - -) < +2 which are dictated by the range of the internal coordinates, and: -2 < <K +2

which follows from the choice of internal force constants. Since: A = A -o 9 m wxo m and A A _n, m Ao, m-n all nonzero submatrices of A near the diagonal ere known if, o,Ao and A - are obtainedo C. LINE GROUP SYMMETRY ANALYSIS An isolated polyethylene molecule in the extended configuration has the one-dimensional Vh space group symmetry.29 There are eight symmetry elements in the Vh line group, as indicated in Fig. 4.3, which are defined in Table III. TABLE III Vh LINE GROUP SYMMETRY ELEMENTS Element Description Tn Pure translation along chain axis of magnitude nc. C2 Rotation about chain axis through angle t, followed by fractional translation of magnitude c/2 along chain. Fvr Mirror reflection in plane normal to C-C-C skeletal plane, followed by fractional translation of magnitude c/2 along chain. av Mirror reflection it skeletal plane. orh Mirror reflection in plane of CH2 group. C2 Two-fold rotation about axis bisecting H-C-H angle. i Inversion operation. C2 Two-fold rotation about axis normal to skeletal plane.

44 i/ CT iCV Li. -- ah C2 Fig. 4.3j. Single-chain polyethylene symmetry operations. Of these eight elements, only T, av, C2, and Uv leave qc invariant, and so these four elements form the star group. The multiplication table for the star group is given in Table IV. TABLE IV STAR GROUP MULTIPLICATION TABLE FOR SINGLE-CHAIN POLYETHYLENE T av C2 Uv T T av C2 av -v — v T av C2 2 C2 C2 v T v av v C2 av T The number of classes in the group, which is equal to the number of irreducible representations, is known if the class to which each element belongs is determined. Elements in the same class are conjugate. That is, the ele

ment R and X R X 1 are in the same class.31 For example, if R is chosen to be av, then all conjugate elements are given by: -1 -l - X1 R X EuvE 1= E1 C2aC2 = C2aC2 C = Cav = v v V vvv = vC2 = av So av is in a class by itself. The same is found to be true for E, C2, and av, so there are four irreducible representations. The representations symmetric to C2 are labelled 32 the A species and those antisymmetric to C2 are labelled the B species. The representations symmetric to av, are labelled with a subscript one and those antisymmetric to av, are given a subscript two. The character table30 for these one-dimensional representations of elements in the [qc] subgroup, the star group, is given in Table V. The character of a matrix operator in any representation is just the trace of the matrix. TABLE V STAR GROUP CHARACTER TABLE FOR SINGLE-CHAIN POLYETHYLENE T'v C, Cv A1 eiqd eiqc/2 eiqc/2 1 A2 eiqd -eiqc/2 eiqc/2 -1 B1 eiqd -eiqc/2 -eiqc/2 1 B2 iqd eiqc/2 _iqc/2 -1

46 The term d is equal to any multiple of the unit cell dimension c, for d = Ic and o < I < N-1 Note that the orthogonality relation is obeyed55 X X*(7)(R)X(7')(R) = h577 R where h = 4. The number of times each irreducible representation occurs in the reducible representation is34: n(y7) = 1 X X R(Y)(R)X(R) (4.5) h where X(R) is the character of the reducible representation for the operation R. The importance of this number, n(7), for the yth representation is that it also represents the number of symmetry coordinates required for that repre.sentation. TABLE VI REDUCIBLE REPRESENTATION CHARACTERS FOR SINGLE-CHAIN POLYETHYLENE X(E) = 6(atoms) x 3 = 18 X(Cv) = O(atoms) x 3 = 0 X(C2) = O(atoms) x 3 = 0 X(ov) = 2(atoms) x (1-1+1) = 2

47 These results, may be inferred from the following comments. The translation operation, T, leaves all displacements for the 6 atoms in a unit cell unchanged,* while cv leaves carbon displacements both in the chain axis direction (z) and in the x direction unchanged but reverses the sign of the carbon y displacements. The operation av exchanges H and H atoms so has zero diagonal elements for the hydrogen atoms. Of course X(av) = X(C2) = 0, since no atoms are left undisturbed by these fractional translations. Therefore, the number of times each irreducible representation appears in the reducible representation is found from Eq. (4.5) to be the following: n(A1) = 1/4 ((1 x 18) + (1 x 2) =5 n(A2) = 1/4 ((1 x 18)- (1 x 2)) 4 n(Bl) 1/4 (18 + 2} 5 n(B2) = 1/4 (18 - 2) 4 D. CARTESIAN SYMMETRY COORDINATES In addition to the 5c phase dependent symmetry coordinates referred to in Chapter 3, which consist of linear combinations of displacements of atoms in adjacent chemical repeat units, additional symmetry coordinates may be written. That is, nine local cartesian symmetry coordinates for the nth chemical repeat unit may be obtained. This is possible because the CH2, unit itself has internal symmetry due to the ov mirror operation. The local symmetry coordinate vector is: Sn = Un(v)Xn (4.6) *Considering a translation T equivalent to E for a single unit cell.

48 The components for the yth irreducible representation are given by35: SiY = nR X47(R)R R The first five rows in Sn for the A1 species are symmetric to av, and the last four rows for the A2 species are antisymmetric to av. i is a generating coordinate, as for example an x, y or z displacement of a hydrogen or carbon atom. The results are shown in Eq. (4.8). Sin = X4 S2n = AZn HI H2 SXn n+Xn Symmetric AYnl AYI2 to S3n = m-. AH1 TH2 to AYn -AYn S4n = 2 a = AZ +AZH2 (4.8) S6n = AY inH1 H2 Sn n An Antisymmetric =7n AyHnl+AyyI 2 to S8n AZH1 -AZH2 Q -' I'

49 The transformation matrix is given in Eq. (4.9), as follows: AXC AYc AZc AxHl AyHl AzHl AXH2 AyH2 AzH2 Sln 1 S2n S3n 1 1 11 Sn IS 8 i/l -1/4f S7n i1 C2 The screw-translation symmetry coordinates may now be obtained from the local coordinates, since X(?)(, ) = X()(R)R sn(ov) (4. o) R for R = E -2 - 37 (2-) 2R 2,c 2,.. 2 where the phase is defined as 5c = qc/2 = G/2. Then X(A) (5)= T1S (v S ( )+eiq/2 S (4.11) — s (b))+ic/ +l V -n+2 (. l) or36 2N-1 (A)(sc) = 1 m e ic Sm(Ov) (4.12) m=O

and 2N-1 )XB)(b,) = L - eimle e c S (av) (4.13) m=O where N is the number of unit cells along the c axis direction. The real and imaginary parts of these symmetry coordinates may be obtained.37 Re ([') (c)] = [Xs ) ] 1 -s -s im)[Xy ) i(c)f =1(4.14) IMIX()X ('?)) - ( I L —S — 6 so 2N-1 Re X(A)( 1)] = (1)/2 cos m5 \"N m=O (4h.5) 1/2 2N-1 m= (hA)' / sin moc Smn(av) m=O w'here 2N = the number of chemical repeat units in the chain. Also 2N-1 Re(x(B)(c)] = L cos m(6c+i)~S (Ov) m=O 2N-1 (4.16) Im(X(B)() = ()/2 sin m(Sc+t)Sm(av) m=O There are now 36 real symmetry coordinates in place of the 18 complex coordinates. The order of these 18 wiLl be altered so that the first nine A

51 species coordinates are symmetric to the C2 operation, which in reciprocal space takes +qc into -qc, and the next nine A species coordinates are antisymmetric to C2. The same is dorm for the 18 real B coordinates. The usefulness of doing this will be made clear in the next section. Nine real A species phase dependent coordinates, which are symmetric to C2, are shown in Eq. (4.17).* 2N-1 XA1 )(Sc) = ( cos m5c m=O X2Al)(3c) m= (N)l ) A4 sin m(c m x(Al)( ) = ((1)1/ j AYX m +Am ) cos m5 A) (6)=(1 )/ (A iyH1 H2m X l (3c) = (l) j + A sin m6c (4.17) m (A2) ) (l) AYc sin msc X6 (sc) = m c Z4 — in m347 (A2) 1/2sin m m (A (/ (AY + )sin H X A2)(5c) = (l X. a M5 9xA2 )(c) = (1) X (Zi-AZ2cos mbc *Recall that Eq. (3.31) or the alternating coordinate systems are used for the phase independent transformation, so no (-l)m terms are shown here for the x and y displacements.

52 The nine A coordinates which are antisymmetric to C2 may be obtained from those above by simply replacing all cos mSc by sin m5c and all sin m5c by -cos m5c. The 18 B coordinates differ from the above coordinates only in a term (-1)m or cos met which appears in each coordinate, since cos m(5c+t) = cos mSc cos ma sin m(6c+c) = sin mbc cos m. E. PHASE DEPENDENT DYNAMICAL MATRICES The 9 x 9 submatrices A of the intramolecular cartesian force constant matrix A may each be factored using the local symmetry coordinate transformation matrices, one of which is given in Eq. (4.9). Then Anmn(Vv) Um(av)Am n(v) (4. 18) where now Al I 5 x5 5 A Gnv) (4.19) I A2 4 x 4 The translational and screw translational symmetry is also utilized to block diagonalize the 18N x 18N cartesian force constant matrix A into 9 x 9 blocks, each of which is factored as shown above. By using the complex symmetry coordinates from Eqs. (4.12) and (4.13) the following is obtained.

'' *B species | I #' /'- bc=+ (s+l) (Jr/N)+ l 5' 1 4 4x4- A species 4x4 b_=+s(t/N) 5 x 5x5'x A L 11 4x4: iB species 18, I I 1 (4.20) There are 2N complex (9 x 9) blocks and O s < (N-1)/2, since the Born cyclic boundary condition requires for iqr = iqn'c = eiGn' = e 2 m =im (4.2l) that the Nth unit cell be in phase with the first, or that for N an even number. The values shown in Eq. (4.20) for 6c in the B species blocks result from the fact that the B species symmetry coordinates differ from those of the A species only in a J phase factor. Also see Fig. 4.4, which follows.

B Species A Species A Species B Species -2 -f O0 r 2 PHASE ANGLE (5 ) Fig. 4.4. Dispersion curve branches for symmetric and anti-symmetric species. Instead of using complex symmetry coordinates however, real coordinates such as those given in Eq. (4.17) for the A species are used. Then each (18 x 18) block for a given 5c value becomes a (36 x 36) block as: 18 - A B l36 (4.22).... i- -— 7 IB' 36 where the primed species are anti-symmetric to C2 and the unprimed species are symmetric to C2. A somewhat different complex to real transformation is discussed in Chapter 7 for polyvinylchloride, since PVC lacks the symmetry operation C2 which simplifies the transformation here.37 The A and A' blocks are identical and the B and B' blocks are also identical in this case. The proof, which is not difficult, will be given now.

For the A' species cartesian symmetry coordinates which are antisymmetric to C2, the phase dependent submatrix of coefficients, U(Ao, () is: A' X1 (5c) sin cz5c X2 (5c) -cos d5c X2-(0c) s XA ( ) sin 0bc ]C>,<c (c Nsin oc5c u(A' )(c) =(1)1/2 x 1' (F)) -Cos aFD X6 (A c) -cos c65C 41 (3, ) -cos arc (A') = (A )S (a) =-Cos - 6c An A' species sinubmatrix on the diagonal of A(c) is shown by Eq. (3.42) to be: A(A(A') (F= U(Al)() S_((Av) A x(A')() = U (AA)'( )A (vA) (4.24) A given A s (n ) matrix element is multiplied by a factor, the value of which depends on the element's row and column indices, as shown below for the A' species. % =:Do::a Q~, =0.D, y+fZ

Column j = 1,3,4,9 j = 2,5,6,798 i = 1,3,4,9 | +sin C(6csin(*+7)5c - X -sin caccos(Q+y)5c a=O C6=0 Row 2N-1 2N-1 i = 2,5,6,7,8 -cos a csin(c+7)c +cos ccOs(a+Y7 )c N L N cx=O j=O For large N: N 7 cos Uccos(a+y)5c = 1 7 ( 2 e (2 2 + cx=O a 2N-1 1 j eiy [e i2a5+l] + eiY [ei2a5+1] = 1 2N[ei'5 +e'i] = cos Y5c 4N The other sums are also reduced, to obtain a factor for the A' species (i,j)th element: Column j = 1,3,4,9 j = 2,5,6,7)8 i = 1,35,4,9 cos Y7c sin y5, Row i = 2,5,6,7=8 -sin y-c cos Y5c for a given A (av). This is identical to the result for the A species, symmetric to C2, which is given on page 59. This result, that the A and A' species blocks are identical, is a simple statement that the dispersion curve (frequency solution for q) is

57 the same for +qc as for -qc. Recall that the C2 operation in reciprocal space takes +qc into -qc9 but as seen above this does not alter the dynamical matrix. Furthermore, elements in the B species block, A(B) (c), differ from those in the A species block only in the Tc phase term which appears in the B species cosine or sine arguments, as cos y(5+Tc) or sin y(5+T). The significance of these observations is that only the A species (9 x 9) (5 x 5) ~ (4 x 4), 3c dependent block need be used in the computer calculations. All B species solutions are obtained by simply allowing Fc~ which has range 0 < 5 < </2, to take on values between </2 and Ai. This unfolding of the dispersion curves is shown in Fig. 4.4. See also page 117, Ref. 31. As already described by Eq. (4.25), each of the phase dependent Ao (oc) submatrices is assembled from a sum of cartesian force constant matrices to which the mass weighting is then applied by the G /2 submatrices according to Eq.(3.41) to obtain the phase dependent dynamical submatrix. An A species phase dependent force constant matrix is (A () U( A)(5c) ( ( )) (4.26) — where o m E. (4.17):c where from Eq. (4.17):

58 SlXe a)( S5, SQ S2~ Sa S47 Sc S6a X2 (5c) sin cr% 3Al (c) cos a~c XA41(c) cos ac UjA)(5c =1 1/2 Al1 =(A, - (1j/) x X2 (,c ) sin a x5 X7 (5c) sin in a X8 (C) sin aa xl2(sc) cos c)J (4.27) Since only diagonal blocks of the block diagonalized A matrix are nonzero, only m = O is needed, and (Gl/ )o AO (A5c ) (Gc )0, C A (3c) (4.28) =C 0.~O~io.,o =C 0.,s ~Sc where (A) ( = X U(A)(6c)A (v y+G(jc) (4.29 since X(2(Sc)5)ym+a O,P+a(, Q ) of C, since Ec takes on 2N value4s.

59 From XU(A) ( )Ao (v)U)(A) (5), 7 in Eq. (4.29), one sees that matrix elements of a given A (av) matrix are multiplied by a sum of terms involving cos (aS) and sin (ao). For a given row and column element of A (av), the A species multiplication factor is: =0,7 Column j = 1,3,4,9 j = 2,5,6,798 21N -I'. i = 1,35949 cos a5 cos(a+y)) -- cos ca sin(ao+y)3 N N Row i = 2,5,67)8 - sin 5 cos(e+y)5 1 sin 55 sin(a+7)3 N L N [i = 13,9459, 9 = 25,6,97,98 For large N, certainly the factor for I = 1 [394 and i = 2,5 667)8 is identical to the corresponding factor shown previously for the A' species on page 56. The {i = 25699 81 factor shown above is also the same as the i = 219 979 Al species factor shown on page 36, but with opposite sign. Therefore the result is that the phase terms to be applied to the A (cv) matrix elements to obtain the A species,.4Ao (5c), block are: Column j =,35,4,9 j = 2,5,6,7,8 i = 15394,9 cos 7yc sin 75c Row i = 295,6,7,8 -sin y7c cos y5c

60 Finally then: i = 1,3,4,9 (A(A)) = (A (v)). cos Y --— oo i,j =o,0y i,j c 2' to = 2,5,6,7,8 (A(Aj) = (A (v))ij sin 7y5c j 6,6,4,9) for -2 < y < +2. In this way, one (5 x 5 ) and one (4 x 4 ) matrix is obtained for each value of b0 for 0 < 3c <. The diagonal Gl/2 submatrices are then applied to the (5 x 5) Al and the (4 x 4) A2 matrices to obtain the dynamical matrices as indicated in Eq. (4.28). The solutions to the block diagonalized dynamical matrix Eq.- (3.25), as rs = rS A = A=C =6 for which each phase dependent diagonal block of C- is given by Eq. (4.28), are displayed in the following section. F. DISPERSION CURVES AND ONE-PHONON FREQUENCY FUNCTIONS The frequency solutions for single-chain normal polyethylene which were obtaihed from Eq. (3.25) are displayed in Fig. 4.5. The labels on branches

61 to v match the notation of Tasumi, Shimanouchi, and Miyazawa.24 The corresponding frequency solutions for single-chain deuterated polyethylene are shown in Fig. 4.6. The labels in the latter figure match those of Tasumi and Krimm for the deuterated crystalline polyethylene results.4 These results match the published frequencies well enough to provide confirmation that the methods used in this analysis are essentially correct. The eigenvectors corresponding to these frequency solutions are used to construct the longitudinal and transverse G-functions for single-chain polyethylene as described by Eqs. (2.1), (2.28), and (2.29). These directional frequency functions are shown in Figs, 417 and 4.8. The force field wiL1 now be augmented -by including intermolecular forces and the modified frequency solutions and eigenvectors will be obtained for a'lattice of extended polyethyLene molecuLes.

62 V1 2800 2600 16oo 1400oo L-' v7 1200 1000 800 - 6oo o00 -- / l9 200 i 0.1.2 ~.3x.4 ~. 67..7r * 97 PHASE ANGLE (Se) Fig. 4.5. Single-chain polyethylene dispersion curves.

63 V1 180 16oc 140 120C c v,~~~~~~~~~~~~~~~~~~ 800 20C v.9 "9~~~~~~~~~~~~~~~~~~~~~9 Fig3~.42.6,,.T [eutr s PITA,-E AITGLE:i (5'IC Fig. 4.6. Deuterated single-chain polyethylene dispersion curves.

100' 10 F 0.025.050.075.100..175.200 50 75.400 PHONON ENERGY (eV) Fig. 4.7. Single-chain polyethylene longitudinal one-phonon G-function, GI (e). — ~~~~~~~\/

10 _ 10 10 I- I C I.02.050.075.100.125.150 17.200.400 PHONONT ETNERGY (eV) Fig. 4.8. Single-chbin polyethylene transverse one-phonon Gfunction, GI(E). ~-~~~~~~~~

CHAPTER 5 CRYSTALLINE POLYETHYLENE The following development owes much to the published works in which the normal mode frequency solutions for crystalline polyethylene are described.39'4~ In this chapter, the methods for calculating both the frequencies and cartesian polarization vectors required for construction of the directional frequency functions of crystalline polyethylene are described in some detail. A. CRYSTAL STRUCTURE AND COORDINATES The polyethylene unit cell is orthorhombic, with dimensions: a = 7.155A b = 4.899A at liquid nitrogen temperature. The unit cell contains four CH2 chemical repeat units in the two molecular chains, as shown in Fig. 5.1. The chain setting angle, G, is assumed to be 480.23 Tasumi and Krimm have studied the dependence of the CH2 frequencies on this angle40 for the range 38~ < Q < 52~. However, their intermolecular force constants which are used in this analysis are determined for Q = 48~. The CH2 units denoted by dashed lines are those at a distance c/2 above or below the units shown with full lines. For bookkeeping purposes, the atoms identified with circled numbers are either in the nth or (n + 1)'st units while the uncircled numbers refer to atoms in the lower (n - 1)'st unit, looking down the chain axes. 66

67 b b -- b -- 2 r r'.' a ____a 2a~~~-1 2~ 2+1 n m-1 Fig. 5.1. Polyethylene crystal structure and intermolecular "Internal" coordinates. Twelve intermolecular hydrogen-hydrogen distances are identified. Again, a, circled distance is for the upper of two distances which have identical projections on the horizontal (a, b) plane. For example, the "internal" intermolecular stretching coordinate, Arl, corresponds to relative displacements of atoms numbered ( and ( along the line joining their centers. The number twelve is dictated by the choice of intermolecular force field which limits the range of the forces considered to r < 3.0A.3940 The coordinate system chosen for each CH2 chemical reneat unit is shown also. Recall that using these alternating cartesian coordinate systems to calculate the Bs matrix elements is equivalent to first calculating B elements and then obtaining Bs = B U (+)t from Eq. (3.31).

68 The individual chemical repeat units are identified by using the l,m,n indices for locations along the respective a,b,c, axes. The relationship between these "internal coordinates" and the cartesian coordinates is: R' = B' X from Eq. (3.35), where the only nonzero submatrices of B' related to the "internal" coordinates of the (0,0,0) chemical repeat unit are: =0,0,0 =1,1,0 =1, 1, B00'0 B00'0 B 0'00 =0,2,1 =1,1,1 =1,1,1 =0,2,10 B0'0' B'0'0 B0'021 B1,1,1 =1,1,0 and these 12 x 9 submatrices of the B' matrix appear in the 0, 0, 0 row as shown below. ~=-1 Q=O ~=+1 m=O m=l m=2 m=-1 m=O m=l m=2 m=-1 m=O m=i B =, J 0,I 0 0 0 I Crl O 4, Or- C 0 r-I 0 0 C04 Ocd Ir —IClr- C 01, —__ _0 0 C 0 O0, —0 — (5.1) The cartesian intermolecular force constants may now be determined.

69 B. INTERMOLECULAR FORCE CONSTANTS The intermolecular forces acting on the hydrogen atoms in the l,m,nth CH2 chemical repeat unit are indicated in Fig. (5.2). nb~~2 m 0 Fig. 5.2. Intermolecular forces on hydrogen atoms in a CH2 unit of crystalline polyethylene. The force constants chosen are those of Tasumi and Krimm40 The forcece nstants chosen are those of Tasumi and Krimm. This results in an "internal coordinate" force constant submatrix'

70 Arl, Ar2 Ar LIA Ar Ar Ar Ar Ar Ar tr Ar ArV1 2 nj -- 4 A r5 6 7 8 r 10 11 12 Ar f 1 1 Lrt fp Ar2 Ar f r4 f4 Ar f2 5 Ar6 000 Ar 880U 7 ar f3 Ar IAr f Ar0 Ar f 12 f 4 (5.2) whereo f.,.0048 md/A f4 f 0205. 4 = at liquid nitrogen temperatureso It is important to also mention that in these crystaline calu.ations5, -the intrarmolecu.lar torsion force constant, Fr,, was taken to be 0,05 mdyne-.A~/rad whereais the value 0~.107 mdyneA /rad2 was used in the singl.e-cha:in calcuLation. An intermolecular force constant matrix in cartesian coordinates is obtained from~

71 C x 9 9 x 12 12 x 12 12 x 9 AO0~O' ~,e n (B' )0)00 F'1P,Y, B'""7 (5.3) A- (Bmn O=, n Ol, mn But cF'- "''3'g'.. F'0,0,0 (5 4) Due to the choice of 12 internal coordinates which "belong" to the (0,0,0) unit; all possible intermol ecular forces f)n hydrogen atoms in unit (0,0,0) are given by F'1010, and OF,1O' 0,1 O., ~ _,GIC1. i, 1 7''' ~. -aC,/,, 0,0,0.for ("YP') e not equal. to zero. Thereforeore A' O - (B')00 F':- O00 B'00,0n (55),mn, =, - 0,0,0-,mn Note that the intermolecular B' submatrices are of order 12 x 9 and the in:termolecular F' force constant submratrices are of order 12 x 1.2. Recall that the int:ramolecular B matrix is partitioned into 10 x 9 blocks and the intramolecular F force constant submatriceS are 10 x 10. In both cases of course the resulting cartesian force constant submatrices of A and A' are of order 9 x 9o C. S^ACE GROUP' SYMMETRY ANALYSIS The factor groub theory analysis of crystaiiner polyethoylene for s 0, ha.s been pu:bliC:hed>[2 The present discussion considers primarily those sym

72 metry operations which leave qc invariant, for all values of qc, when~ -/c < qc < t/c. The polyethylene crystal has P (21 1 1) space group sy:in..try, which is n a, m g also referred to a.s P nam symmetry~ It is also referrefd to as D26 kpace group symmetry in the Schoenflies notation, The symbols have the following meanings in the Herman MauEin notation43 P - Prirritivc lattice 1, 2 refeis to the order of the screw-rotation axis in thi a dirbction. n Subscript I divided by 2, or 1/2, refers to the f'racticn o-' t-e- unit cell over which the translation occurs. Sym.bol n refers to a, diagonal glide plane perpendicular to the C oxis with a translation cf magnitude (b + c),/'2 2, a- - Symbol 21 refers to a, twofold screw-irotati-on axis in the b cdirection with a. fractional translatlon of' 1i'L x b0 Sy nbol a indicates a glide plance perpendicular to the b axis wiith a fractioial translation of inagnitud, a, 2~ 21 m - Symbol 21 refers to a, twofold scl-w-rstation axis along tle c -irection wit'h a firacticnal translation i,' ma,gritude c/2. Symbol m rer'ins to.-a mirror plan._. perpendicular to ti a c a.,isO Those operations in the subgroup of tiais space group which leave q, invariant are shown in Figo 5~ 3. This subgroup is referred to as the star group or (qc)'ave vector group)o0

73 b v(bC ) -.. _ I (ac) a Fig. 5.3. Star group symmetry operations for crystalline polyethylene. These operations are identified in Table VII. TABLE VII STAR SPACE GROUP ELEMENTS FOR CRYSTALLINE POLYETHYLENE T - Translation by an integer multiple of a, unit cell dimension. CTv(ac) - Mirror reflection in an a.c plane, followed by a fractional translation of magnitude a/2 along the a axis direction. %v(bc) - Mirror reflection in the bc plane, followed by a, glide of (b/2 + c/2) in the bc plane. C2(c) - Rotation about the c axis through a chain, followed by a glide of magnitude c/2 along the c axis direction.

74 In addition to these four elements, the ah mirror in a, CH2 plane is also shown in Fig. 5.3. The latter element is used in obtaining real symmetry coordinates from the complex symmetry coordinates. This element takes +qc into -q., which does not alter the frequency solutions. This is seen in the symmetry of the dispersion curves about qc = 0. The number of nonequivalent irreducible representations is equal to the number of classes in the star group, (qc), while elements in the same class are conjugate to each other. In an abelian group, every element forms a. class by itself and has a one dimensional representation. Keeping this in mind, the multiplication table is formed and the number of classes is determined. TABLE VI-II STAR SPACE GROUP MULTIPLICATION TABLE FOR CRYSTALLINE POLYETHYLENE Tc av(ac) av(bc) C2(c) Tc Tc Zv( ac) v(bc) 72(c) a (ac) v(ac) Tc C2(c) av(bc) <v(bc) Ev( bc) C2(c) Tc av(ac) C2(c) C 2(c) av(bc) av(ac) Tc Elements in a. class with av(ac) are: E v(ac) E-I = J v(ac) -v(bc) Tv(ac) dv (bc) = jv(bc) 7v(ac),v(bc) = v(bc) C2(c) = ~v(ac) 2(c v(c) 2(C) =v(ac) 52(c) = a2(c) v(bc) - (ac) ",<~~c, ~~~~~~~~'; ~

75 Therefo:re 5av(a,c) forms a, class by itseifo The same is true of av(bc) and Thr02;)(c)and There are therefore four one-dimensional irreducible representations in the -,c ] subgroup of P:nami Those irreducible representations (species) which are symmetric to C2(c) are labelled the A species jand those which are antisymmetric to C2(c) are label led the B species. Those symmetric to the'av(ac) mirror are label.led with a subseript 1. and those anti-symmetric to?v,(ac) are labelled with a, subscript The character talble fo:r these representations, when qa - qb:Q O is~ TABLE IX STAR GROUP CHARACTER TABLE FOR CRYSTALLINE POLYETHYLENE e aJv ac) cJv(bc) c) A- e.iqn'c +i + eiqc/ + eiqc/2 A2 eiqn -1 - eiqc/2 + eiqc/2 B3 e i Qnc +1 eiqc/2 iqc/2.IBe.|e iqn'c e iqC/2 eiac/2 qiq 2n nc 2 ine Irn T albove, the character is e ieq nc: e e e where n 1 a,: 1s the ehe..ical repeat units, n' label s the unit cells, and qc/2. Note that the character X [C2(c)j in the B representations is ei I, ei''/ ) which is the product of ein and the character for the A re'.re!-entra.t i. no This rela.tionship will. be used later to simpl.ify the procedure:Cor Y 0, ata ini ng the dynamical matrices in *the B representationso

76 Note also that the characters for the 7th representation obey the orthogonality rel.at ion.33o X I*(7)(R) (7 )(R) 4~r (5)6) R In orde:r to determine the number of symmetry coordinates required for each representation; it is necesDsary to know the number of times each irreducible.rep~resentation appeasrs in the reducible representation for a symmetry element. 34 This number is.. (?,) - 2 *(7)(R) X(R) (5,7) R where,X(R) is the character in the reducible representation. X(E) = 56, as there are 12. a.toms in a, unit ceil. and x(/v(ac)) = X(v(bc)) = ((c)) = O, since these operations..:eave no atom displacements unchanged.. The number of times each irreducible representation a.ppears therefore iso (A, ) _I n(A) -- L [1 x.6j 6 - 9 n(A2) 9 n(B.) = 9.(BA). 9 9 This means that each irreducible representation requires 9 symmetry coordinates and that 9 x 9 dynamical. ma.trices will a.ppear in the block diagona.lized dynamica.l mat:r ix o

77 D. CARTESIAN SYMMETRY COORDINATES Since there are four one dimensional irreducible representations, each of which appears nine times in the reducible representation, there must be four sets of symmetry coordinates. Each set contains nine components. A given component or symmetry coordinate may be obtained from35: S(7) = X(R) R (5.8) R where the sum is taken over all symmetry operations, R, of the subgroup of the 21 21 21 space group P n a m - which leave the phonon wave vector qc invariant. Recall that only the phase shifts along the qc axis are being considered here. By operating on each of the cartesian displacement coordinates of the zeroeth CH2 repeat unit atoms and applying the appropriate character, a set of symmetry coordinates may be obtained. The symmetry operations are: R = E, C2 (c), v(ac), v(bc); (2c), 2 v(ac) 2(c), which includes all products of the four elements of the (qc) group and the translational subgroup of the space group. In this way, the first nine coordinates, which are labelled the A1 species to indicate that they are symmetric to both the C2(c) and the av(ac) operations, are found to be: 2Na-1,2Nb -12Nc-1 S(A )(c) = z (1)n= Aamn einSc s(A c) = mn in6c =(1m Abc m e

78 (A1) (e cnn;2__ -,mn H S4/ i tbc = Aam( n Ace e innc (A k -1 BH in5 S1. 6 ( -= \ (-1) Ab e c imn ( A c.) = I Ln e H Qmn i,,)'r QACH n5r:. alzati+n H2factor is = (, ere is the n toe nuu~iber of unit 0113 in the a, b, a=nd c direotion3 f nowi the char>actor- for the A2 specIes are us K., tche nine couriznates 8 A,.:in (A H2 in5 o 2 {5c,,! = \ (1)dm inr > _ IL 2)>t _, C c -J6 Rmn 2r) 1 / )rtialization fa-1ctor is B. = (NjY..cr of ab( ab/ -,'.:~,e t; ru:its in the crysjtal. 11- 2IN 2 1N 2N, ~f~r3 IN Nb~T a. rcs p:...~ctvely'-..... a, C, a C the~. lnwbt.er if unit cells in the a, b, aind c directions~ f now the characters for +,tLe A,2 species are us-.d the nine coordna'-es wI":J_~h are sletric tJ r2c) but Cantisymmetric to 3via c) are found to b3e tb... ~inn A2A7 r-, in rin~~~~~~.~l 1_..-

79 (A 2. (5c) 7(-1)n AbH einbc Qmn 5A H inb rmn (A2) ) T ( 1)m AcH1 e cn QImnRmn S(A2 \ + 2 in 27 sc " -1 Aa e c (2)(6 ) l(zl)n.b mn in imn s(A2)( ) IZ(-.)m H2 einbc Imn The nine B. species coordinates which are antisymmetric to C2(c), but symmetric to:''v(ac) are. S(B1) (65) = Aa eincmn Qmn Anmn S(B1) (6c) c ()n A b n in6 Q mn: (6C) 7 (dl)m AbH' einSc 1m Lmn 4 ((5C) = eib in6c ~mn (B1) () a ein6c S(B1) () l)m Ab mn einSc imn (B H'1 in~(.mn (B H-) H 98 (be) d) Acr mn e 6c L ran5

80 The nine BS species coordinates which are antisymmetric to both C2(c) and Th(ac) are: S(B 5e) - ~ ( 1)m Aamn einMc ^tB)(" ~ ( nAb rnn e in6 ~(Bn)( 5 ) c H in6 2 C Arnenc (B3' (.)m -n mn ea in6 2?O ( _PX(i) A e C 4 2rn ^ "(5) _: q (AH~A Aa ein6 ~mn'1 ain (B5 ) ) n ein C m= 5 r.mn IL in6 1 rrm mn -7- -I rrn'4- H2 e ~ mn Notice thaet ea'h terrr in the surm_ for a given B1 symm.etry coo:rdinate differs from the cor:responding term in an A, coordinate only by a factor ein, (l)n Each B. symmetry coordinate also diffe.rs from the cor:responding A0 coordina.te by m,,y be cbta~i:ned. In vector nota,tinnc thL~~~~~~~~,O fat_~.Thsr'toshp-i- Abe u.e;-Dusruntyogodavng No-w the rea.11. ad imaginary parts -)f the thirty-sixcomplex coordinate

Re!S? (6)) 9 [s(~c) + S())(+ Im, (~C0-) 2 [S( )(bC) s(C)(6()* (513 ) i.n whi. ch ca.se' ReS( )C e-cin c S(Y) cos nb' — c- — ~mn Qmn ImiS(Y)( c)) 2 j einSc S(W) sin n5c (5.14) rim — ~mn f AY -A A20 For the B species coordinates: Re. (B)(6c) v-: e-inScS(Aj) cos (6e t r)n ~mn Im[S(Sj)(~c)] = i r \T e-in6c S(Aj)sin (6c + n)n (5.15) - Q m n where j =:.,o There are now (2 x 36) real. symmetry coordinates in place of the original 536 com.plex coordinates Further use of the crysta.l symmetry may be made by ut-il izing the horizontal mirror plane Ch which pa.sses through the CH2 repeat units. Theorder of the AI and A2 coordinates (36 in number) is rearranged so that the first nine a.re A1 coordinates symmetric to 9h' the second nine are A2 coo:rdinates'symmetric to &h; the third nine are A1 coordinates antisymmetric t,-:h and the La.st nine a:re A2 coordinates antisymmetric to oho The same re-,a:ra.ngement is performed for the 36 B species coordinates.

82 Use of these seventy-two real phase dependent coordinates makes it possible to obtain the block. dia.gonalized real phase dependent dynamical matrix so that, for each 6c value, the following set of 9 x 9 submatrices is obtained: 72 Al A 4 CS(5c) B72 Real B2 BThe primed matrices are those which have coodinates antisymmetric to ah (.5o:.6) The primed matrices are those which have coordinates antisymmetric to aho The real symmetry coordinates which make this factorization possible are: sA1")( ():.a, i,[('l)n] Aac cos nb8 C Imnm S(Al.)(Sc) ( — 2, (..)m+n mn cos nbc UA(AC s imn s i n n m mn c S(hA.i)(6c) (-)n ] c S$A1)(6c) - & n,[(.i)m] Aba mn cos nbc m rmn

83 S6 (6c) 2 l A H1mn sin nbc ~mn S (A,>l)(6C = a [(_l)n] (Aa'H2 cos nb 7 mmn S(A )(c) = \2 r [(i)m+n] Abm cos nb6 mn (A) —-I H2 S9 (~c) = 2 Ac ~mn sin nb6 (5.17) The terms shown in brackets [] are diagonal elements of [U(Al) (+) ]mn, where: = - - ~mn s(A1)(5c) u(A)(_ ) U(A1)(+) X as described in chapter 3. The A2 species coordinates are' mcn c S(2) (5 Aa, mn cos nSc L mn S(A2) (c) 2 r ()l) Abmn cos nbc ~mn c Q mn S3 (e) ( )2 () c sin nbcc Qmn S 6m+n H1 2 (6c) 2 (n -.l)n Aa mn cos nbc ~ mn c m~mn cos nn ~mn s(A2)(5c) H2 s (l)m.b s nn6 5(A-) m H2 (bc) i12 (4) A~ sn n bc ~mn mn (A2)( ) 2 )mc AaH sin nnbH AS9 (6c) 2 1 (.1)18) l rAm k

The A1 and A2 symmetry coordinates, which are antisymmetric to the ah mirror, are obtained from Eqs. (5,17) and (5.1.8) by simply replacing cos n5c'by sin nbc and by replacing sin nbc by - cos nbco E. PHASE DEPENDENT DYNAMICAL MATRICES The transformation of the cartesian coordinate force constant matrix, A, to real space group symmetry coordinates results in a block diagonal matrix for each bc value as shown in the previous section According to Egq (3.42), a. given (9 x 9) A1 species block is obtained from: (Al)COo. (A)000o (A1)a0r (A) aC P y A (So)ooo a~gy;a' B' r -- a~C)OI~ =, I ~ I A/ooo A (AL)000 U( ~ = CQOe)O' -x,' —, Y' (Xi i' nf an a~nd. 0-Ci + (X P + = Y+; )0,00

As in the single-chain polyethylene analysis, the matrix elements of a 0,0,0 given A =, matrix are multiplied by a sum of terms as shown below, since the transformation subma~trix is: (A )cS S1 1 c os~c S(A1) co s~5c S(A1) S( Al) sinTyc S(A1) |~ (A1)~ ~ 2 1/2 (A)CS - 6,)007 Nabe 5 S(A1) |sin7b s~A1)|c H1 H.1 H1 H2 H2 9 a C b Ac a Ab c H2a b H2 Ac 007' O 00 7Y OOy OO0 OO0 007 007 007 (5. 20) Reca.ll tha.t the (-l)n or (-l)m+n terms in the symmetry coordinates a,re introduced into the analysis by selecting coordinate systems for the different CH2 units appropriately as shown in Fig. 5.1. This is effectively accomplished by changing the signs of B' matrix elements in a corresponding manner to obtain the B' s matrix elements. According to Eq. (5.19) and (5.20), a, matrix element in a, given row and.000 column of A, is multiplied by a factor shown below:

86 0)f,0 Column of A i0'0' - j = 1,2,k4,5,7,b j = 3,6,9 i = 1,2:4,5,7,8 COSY6 cos(y+y" )bc N cosy, sin(y+Y )b Row of 7 7 7 Although the row-column indices here are different than in the single-chain polyethylene case, the sums are nearly the same and the results for the factors a-re ~ Column j = 1,2,4,5,7,8 j = 3,6,9 i - 1.2.4,5,7,8 cos yC 5c sin 7 5c Row i - 3 6,9 -sin 7"c cos 7"5c Therefore the matrix elements of A(Sc) required to obtain the dynamical sub(Al) matrix, Gc/2 A,o>000 /2 - C_(), are -.atrixg G1/9 A()0,0, C0 G1/2 (A&i,2,4,5,7,8 j ) (,2,4,5,7,8 (A ~88 i j-= (A 3 Q) cos yc; for (A )iijj 619 CIBaE~~~~~~Y~ ~(5.21) (A(O Z (A )i sin Yb; for f12,45,7,8 (i 7( 80 0,-(A )J sin y5; for i =;,,9,2,4,5,7,8 As was true for the single chain, the A1 and Al species (9 x 9) blocks are identical and the A2 and A2 species (9 x 9) blocks are identical. As was also

87 true for the single chain, B species blocks differ from the A species blocks only in the factor cos y7T Therefore only two (9 x 9) matrices need be obtained (A1 and A2 species) for ea.ch 6c value. By again allowing 6c to take on values, r/2 < 6c < I, the B1 and B2 species frequency solutions are obtained. Fo DISPERSION CURVES AND POLARIZATION VECTORS Only the low frequency solutions for normal crystalline polyethylene are displayed in Fig. 5.4 as a, function of the phase shift, 6c' between adjacent chemical repeat units. The corresponding dispersion relations for deuterated polyethylene are given in Fig. 5.5. The higher frequency solutions which were obtained differ in no essential respect from the published resultso39'40 In Figs, 5.6 and 5.7 the squares of the hydrogen and carbon polarization vectors for normal crystalline polyethylene are plotted vs. 6c for the vg and v5 branches. For these curves: ab2 H1 + 7 b2H1 aY2H2 +'bH2(H b)"2 ]+ [ ) 2 2 2 H27 1 /2H + y2H2} (ab), 1 f yc2 + c>2 } 2 a. (eQ. 2 In adtition, the polarization vectors for deuterated crystalline polyethylene a!re displ.a,yed vs. 76 in the Appendix. lrit is importdant to point out that the required normal.ization for the polariza~tion vectors, which is indicated in Eq. (2.5 3), is such that the sum

goo 800 ~ V8oo1 700 7oo 600 v p oo? 0. 23.ni.67-.8.t. 9 PHASE ANGLE (Se) Fig. 5.4. Crystalline polyethylene low frequency dispersion curves (FT =.05 mdyne - A/rad2).

8C) - 0C0C - o0.4Pi ~ 7j.8-.~ 9 PTASE ANGLE (c ) Fig. 5.5. Deuterated crystalline polyethylene. low frequency dispersion curves (FT =.05 mdyne - A/rad2).

~ost- Vc~~~~~~~~~~~~~~~ - (rFT~~~~~~~~~~~1- )V5 (YH )2 ab ab V9 (XH)2 C C -9 (HI )2 5 ab t ab 040 ab.01 0 V7~ab.2rsa PHASE ANGLE Fig. 5.6. Squares Of the Polarization vector magnitude5 for 5 and v9 hydrogen motions in normal crYstalline polyethylene vs. phase angle (5c).

9C (0yc)2.2 Cv9 (C )2 ab 7alb C C~~~~~~~~~~~~~~ Fig. 5.7. Square of the polarizat on e to r magnitu for v5 and carbonVmotions in normal crystalline polyethylene vs. phase angle.2~ ~ ~ ~ ~ ~~~~~~.n.... x~"t-~~~~~~~~~~~~HSEA(I C~~~~.1,Li * \*S ua e fth o aizt o ec o a ni u e o ca b n m t o si o m l c y t l in o y t y e e -s h s n l

92 on s is taken over all atoms in a, unit cell. However, in the computer calculations the eigenvector solutions to the equations of motion are obtained from the (9 x 9) A1 and A2 species and the B1 and B2 species dynamical matrices. For each of these four dynamical matrices the eigenvectors are normalized to unity. Instead of obtaining the eigenvectors from one 36 x 36 matrix at ea~ch q value so that the total sum of squares for 36 polarization vectors is 1, the sum of squares from four nine component vectors is 4 at each q value. Therefore, the computed polarization vectors shown in the Appendix for deuterated crystalline polyethylene must be adjusted by a, factor of 1/4 to obtain the desired normalization. The polarization vectors shown in Figs. 5.6 and 5.7 are already renormalized in this way. G. Oi]E-PHONON FREQUENCY FUNCTIONS The transverse and longitudinal one-phonon hydrogen frequency functions are shown in Figs. 5.8 and 5.9,, respectively, for normal crystalline polyetnyl2ne. In Figs. 5.o10 and 5.11', the transverse and longitudinal G-functions are shown for a, deuterium atom in deutera~ted crystalline polyethylene. The "incoherent approximation" is invoked in plotting the frequency functions for deuterated polyethylene in this way, since (CD2)n scatters neutrons primarily in coherent events. In addition to the arguments presented in Chapter 2 to just fy this approximation, this approach is further justified by the very close similarity between experimental frequency spectra obtained for normal and deuterate polyethiylene. 9

93 H. TWO-PHONONJ FREQUENCY FUNCTIONS The temperature dependent two-phonon transverse and longitudinal hydrogen frequency functions for normal crystalline poljyethylene are shown in Figs.,5.12, and 5.13, respectively. Only the energy ranj._ from zero to 0.1 ev is shown. These functions ar_ obtained from the relations for I(E) and GII(E) in Eq. (2. 9). I. DIRECTIOI2AL DEBYE-WALLTR FACTORS AND WEIGHTED FREIJjLICY FUNCTIONS The transverse and icr:;.;itudinal Debye-Waller coefticients are first calculated using Eqso (2o45) and (2025), respectivclyo. n th-se calculations, the frequency spectra are given a, qua~dra~tic fresquency c`Lpurdence below.006 eV, in accord with specific hzat requirements50 and ii. order to gu.rantte a. finite Debye-Waller coefficient and,.on-zero cross-sectioi.. The norhnal polyethylene results for 2W,/-2, on tle temnperature range from OCK to 600~K, are shownr in Figo 5.14. For the speci.l case when neutrons are scattered thrcugslh "'0~ with a.n initial and final neutron ernrgS of.030 eV, the transverse ar.d loilnitudinal Debye-Waller factors for normal polyethylene are as shown in Figo o.15, for the same temperature range. The ratios of e'2WL/e-2WT are given in Fit;. o16 o i st arl.. the calculated ra;tios between the longitudinal and transverso elastic maiitudes, each of which is givern by Eqo (2.19). The relative ma;-nitudes of the transverse one- and two-pnorlor contributions, as a function of tempurature, are seen quite clearly in Figo )oi1[. At the higher temperatures, for example, the height of the.022 eV pea' relative to

100 80 V8 (7T) +20- a LV(O) V/ (0)2 ILJJI LIJ I0 b2" V Z5 (0) 6Q4 V5 (7r) CLh~a V7(0) 1/7 n 2 LIJ v(7r) V (0) 8 M 01.0 Lu.8 F.-6.1 I LU.4-.2 -2 0.025.050.075.100.125.150.175.200.350.375 400 PHONON ENERGY (EV) Fig. 5.5. Transverse one-phonon hydrogen G-function, GI(E), for normal crystalline polyethylene.

v7(7T):00 80 60> 40 IIw~~~~~~~~~~~ v~~~~~~18(o ) >- y3(T7) 1- 20 y( 20 - IV~~~~~~~~~~~~~~~~~~~~~v (7T) LLI 8 1 /5 Z8~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~/ (7)/5 1/9 1/7(0) B6.4 -j.2-.I /1 I I / I I I 0.025.050.075.100.125.150.175.200 35.375 PHONON ENERGY (EV) Fig. 5.9. Longitudinal One-phonon hydrogen G-function, (E), for normal crystalline polyethylene. o.4nomlcrsalnepletyee

100 80 — 60 T2 (771 > 40 V LII - _0 2'9 ( —- 7/(0 - b o V7, (o) Z 75(77 LLI 2I0~ C) Io -- 8z u3(0) 06 a 0 4 l/5(7r) CL 2 LUJ 1..8 I 6 c.4 D -- V ~~~~~~~~~~~~~~~~~~74 (7r1 F — < 1 0.025.050.075.100.125.150.175.200.225.250.275.300 PHONON ENERGY (EV) Fig. 5.10. Transverse one-phonon deuterium G-function, G(E), for deuterated crystalline polyethylene.

I00 60 - v7(T7r) > 40/3 (- 3(o) V7(0) Li 13 (7r) 2/8 (0) >_ ~~~~ v~~~~~~~~~co,~~~~z/(' - Z/q~~4 (0) ~20- / (.f) z l2:2/ I0 82: 7v(0) V4(T o 2 0 4 F(1.0 -8 D F - 0.025.050.075.100.125.150.175.200.225.250.275.300 PHONON ENERGY (EV) Fig. 5.11. Longitudinal one-phonon deuterium G-function, for deuterated crystalline polyethylene.

25 2( H 402 OK402 ~~~0.0.2.3 o4 os.o6.0.o102 K fF 2 E20 T102 ~K ___1022 IK 0.01.02.05.04.05.o6.07.o8.09.0o ENERGY TRANSFER (eV) Fig. 5.12. Transverse two-phonon G-function, GI((E), for normal crystalline polyethylene as a function of temperature.

5 4 402 K O3 H kO 302~K P-L 0.020 F 2 02.K;II 102 OK10 ENERGY TRANSFER (eV) Fig. 5.15. Longitudinal two-phonon G.-f'unction, GII(e), f'or normal crystalline polyethylene as a f'unction of' temperature. r'~~~~~~2O 0.01.02.0j o4.05 o6.07 o08.09.l ENERGY TRANSFER (eV) Fig. ~17~Longitudinal two-phonon G-functiox, (, If,,ra L crystalline polyethylene as a function of temperature.

100 the two-phonon.044 eV peak is decreased considerably. The longitudinal Debye-Waller weighted one- and two-phonon frequency functions are shown in Fig. 5.18. The scale of Fig. 5.18 is expanded relative to that used for the transverse functions in Fig, 5.17, yet the temperature variations are less dramatic. For deuterated polyethylene, the Debye-Waller coefficients at 93~K are calculated to be: 2 02 2WL/~ =.0166 A 2WT/~a - o O06 2 2WT/~~ ~-.0206 U These ma~y be compared to the values for normal polyethylene at 935K, which are: 2W,2 o0205 Ao 2WT/ -.0255 A2 The cone- and two-phonon directional frequency functions for deuterated crystalline polyethylene are shown in Figs. 5.19 and 5.20. These functions are also weighted by the Debye-Waller factors, which are calculated for Ef -.030 ev. and Q = /2.

.10 Transverse.o08 % /0 C\J o6 3I L Longitudinal O.02 o 10 200 300 4oo 500 6oo TEMPERATURE (KO) Fig. 5.14. Calculated directional Debye-Waller coefficients as a function of temperature, for crystalline polyethylene.

1.0.8 cJ 6 ~~ I H Longitudinal.2 ~ ~Transverse 0 100 200 00 400 500 600oo TEMPERATIFRE (K ) Fig. 5.15. Calculated directional Debye-Waller factors as a function of temperature, for crystalline polyethylene (Ei = Ef =.030 eV; as - /2).

F-I 2.0.5 0 100 200 500 400 500 600 TEMPERATURE (OK) Fig. 5.16. Calculated longitudinal to transverse elastic peak ratios as a function of temperature, for crystalline polyethylene (Ei = Ef =.030 eV; GS = /2).

20K 12 11 10 102 ~ X 2 OK 9L. 8 20K 7O ~~~~~~~~02 ~~~~~~~01102 0 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~1020K6 1- 020oK 02 ~~~~~~~~~~OA 3 H 1 0~~~~~~~~0 L. I o02~ Lb 2020o 2o 202. 2020 2 Li 20 -— I 20 I02&220VL2 12K 1~~~~~ 2C200l2220o. —.3O2 5 LF 02 0K 102" C~~~~~~ 02" L~ —-L~_ KK02 oK3 ~ —-~! IE( I Io I0~~~~~~~~~2" 0.01.02.03.04.oo.06.07.08.09.10 ENERGY TRANSFER (eV) Fig. 5.17. Sum of transverse one- anc two-phonon frequency functions for crystalline polyethylene weighted by the Debye-Waller factor (E. = Ef =.030 eV; Qs = i/2, T = 930K).

~~~~~~~~~5~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 20K 4 102OK H w3 202N OK OI~~~~~~~~O -0 202~ 20 0. ~01.02.03 o4.o6<.02 8..lo ENERGY TRANSFER (eV) Fig. 5.18. Sum of longitudinal one- and two-phonon frequency functions for crystalline polyethylene weighted by the Debye-Waller factor (Ei = Ef =.030 eV; Q0 = T/2 T = 930K).

106 45 40 >(1)35 ~30 0 25 ~20 LU 15 eLf U. 0.010.020.030.040.050.060.070.080 PHONON ENERGY (EV) Fig. 5.19. Sum of transverse one- and two-phonon frequency functions for deuterated crystalline polyethylene weighted by the Debye-Waller factor (Ef =.030 eV; Qs = i/2, T = 930K).

107 45 40 & 35 z LUl c 30 z 0 25 T20 J I15 < 10 0I 51 0.010.020.030.040.050.060.070 PHONON ENERGY (EV) Fig. 5.20. Sum of one- and two-phonon frequency functions for unoriented deuterated crystalline polyethylene weighted by the Debye-Waller factor (Ef =.030 eV; Qs = It/2, T = 930K).

CHAPTER 6 POLYVINYLCHLORIDE A. MOLECULAR STRUCTURE AND COORDINATES The crystalline polyvinylchloride chain consists of (CH2CHCL) units joined to form a. planar zig-zag structure with chlorine atoms in adjacent repeat units located on opposite sides of this plane. 4445 This syndiotactic conformation requires two chemical repeat units along the chain direction per unit cell, as shown in Fig. 6.1. C~~n+l H2n +1 Cln+l 1 H3n+l y' ~ r2. W2 1 1 ~C2n 2 4 1 L' HlI r H3 n-l C2nH2n1 Fig. 6.1. Syndiotactic polyvinylchloride chain structure and internal coordinates. 108

109 Polyvinylchloride crystallizes with orthorhombic unit cells of dimension a, = 10.6 +.1A and b = 5.4 +.1, each containing 4 molecular chains.46 However this analysis considers only an isolated chain, on the assumption that the intermolecular forces are significantly weaker than the intramolecular forces. The unit cell dimension along the chain is c = 5.1A46 and the interatomic bond lengths and angles are as given in Table X. TABLE X POLYVINYLCHLORIDE REPEAT UNIT DIMENSIONS45 r c = 1.0o93A c-H r = 1.54A C-C r = 1.798A c-cl Angles = 109~ 28' These are the same as Schachtschneider and Snyders' polyethylene dimensions28 except for the carbon-chlorine bond distance. Using Opaskar's notation, the internal coordinates for each chemical repeat unit are described in Table XI. One of the 20 x 18 submatrices of the partitioned Bs matrix is shown in Eq. (6.1). Only the row and column labels are shown, to aid in interpreting the computer output matrices in which the actual Bs matrix elements are displayed. These matrices are given in the Appendix. The calculation of these elements proceeds as described previously, using Bs = B_ U (+)t where in this case the equivalent alternating cartesian coordinate systems are as shown in Fig. 6.2. Notice that these are not right-handed systems.

110 TABLE XI POLYIINYLCHLOhIDE CHEMICAL REPEAT U1NiT INTERNAL 2O0R'JI.ATES R1 Cin - C2n Strmtc.' l1 Cnr - Cl n Strtch. r3 C1n - H3n Str t' 1I C2nl -C1i - C2n: enA 01 H3n - Cln -C2.n L d -1 C= Q CCl1 -C2r1 Lr n. H3n - C1n C2n-1 Bnd -~ ~C.n Cln - C2n1 2 1H3n CIn - C2n - nd T1 Cili - C2n Tors cr. R2 iln - 1ln+1 S-tretLh r1 2n -Hn Htrlt r2 C2r - H2r Str,t&! CirC2. -Cll ln1 CL+ erLd 51 t~l~hi C2n- H2n -hrnid C11 - C2 - Hi 1nd Y Cl n 2n n nd "Y2 C1l - C2n - H2n B nd 7 1 Cl,+l - C2n - H1n Pend Y'2 Cin+i - C2n - H2n iLcnd T2 C2: - Cln+ lors.or

111 Xn+l n+l Z+1 Xn+l Yn+lZn+l n+l Zn+l Xn+l Y n +l n+L + Z- n+1 X n+1 Zn+1 AR1 AX1 Ar5 3 AG A= AT1 AR2 Ar1 Ar2 Aw2 A?1 An2 A'2 T 12'ST2

112 z Hln+l H2n+l C Qn+l y ---— x n+l H3n+l _ H2n y No x n HHln Cen z H2n-1 x n-1 y Fig. 6.2. Polyvinylchloride repeat unit coordinate systems. As was true also in the polyethylene calculations, the torsion coordinate:26'27 AT = AT i/9 (6.2) i is used:

B. INI'RAMOLECULAR FORCE COLSiTAWI-'S With the exception of the diagonal torsion foo-.ce constant, all internal coordinate constants are taken from Opaskar' s work>4) His set of force constants is a modified version of Schachtschncicdr anr Snyders' coaniolation for the saturatad hydrocarbons, containing in acdditionl. sv-.ral ncwly detrtmirlned force constants The diagonal torsion force constant, FT - ~6 MLynre A/rad2, has also been included in this analysiso This value is chos.. afit r assuming that the torsion force consta.nt for polyvinylcliloride should not b~ cc a,tl- diff rent from that of polye t).ilylene Th.e_ arra-ys of intornal coordinate force coarst..nt natrix el e mnts ar*. given in the Apotendix One of these submatrices is la'.elled F, whic'h is F, for forces onr theL zeromth chemical rmpf at unit fromn irnaL coordirinat displa.c:rments in unit %(1) o The other submatrix is labell-d F3, which is,- for for}cs on atom-s inr thn. zeromth chermrical r(eeat unit fkroi &isplaciirLMnts of atoomis in t:is unit;o Of course, forces on the zeroeth unit;uL-. to d'isplacentents in unit (+1).i are given b~y F01o0 Displacements of atoms in c1lls beyond n = +1 are arc assum.d in this..liodel to produc_ zero force on atoms in th' zcroetfi unit. For each chemical repeat uLinit (CH2CKCL) ccrtairnini six atoas, a 20 x 2J internal coordinat_ force constant submatrix is utilizcd, with row and coluian labels for thle internal coordin.ates as indicated in Eqo (o53)O The cartesian force constant matrices a.r_ obtaintid by again using Eqo

114 A t a I A=o, m X ) y Bo m-Y o The limits on the indices is this case however are: -1 > m < + -1 > 7 <+ -1 >a <+ 1 -1 (m-7-a) < + 1 Only the A and A0 1 submatrices must be determined, since all others are then known from: AO,m =,o and: A = A -n,m =O,m-n C. LINE GROUP SYMMETRY ANALYSIS The extended syndiotactic polyvinylchloride chain has one-dimensional space group C2v symmetry.44 There are four elements in the group as shown in Fig. 6.3. These operations are described in Table XII. C / H H /lc olo4 H C:~,o Fig. 6.15. Single-chain polyvinylchloride symmetry operations.

115 Unit m X1 r3. 1 1 T1: 1 R2 rj r2 c 01 2 ~l 72 1 7 2 X1 r3 ~1 1l /1 P2 72 7'1 7'2 (6.3)

TAPILE XII POLYVINYLCHLORIDE LINE GROUP SY, 1i' " iYLLL EJTS ~~~El','~m~ ~.rl~~~~~jt Dsc:rip' oni4 pulre tran.slation along the ci.ain exis of magnitude nc. C2 twofold rotation about an a.xis t.-,'-ough a CH2 urit, perpendicular to the chrain axis. jv mirror reflection in the skel'-;al pla:-e -'ollcowed by a fractlio!.a.l transla.tiorn of magnitud < c/'' lor.g the c.Lamin. c4h mirror reflection in the hori2.orntal [lane tuirough a. H2CC gr oup. Only the operations Tn and c v leave the m.,gnitud,;;' tif ctt.on ot' qf unulhihimnged in reciprocal space. The operations C-2 and cl ta: +q into -c-c. Since the dispersioun curves are syaimetr-c about q =-, tl-..nts ould also be used in -;riting synoiletry coordinates for block:..,601x l. zilg the'y!l.anical rmiatrices. However only th. q- qJ) sta.l' group elemerts a-r' usdc r.!re sir-e the correct orientatlons for the repeat unit cartc-sian cuor i rr..t t, sst t.ns arei cu penernt upon navirn, subgroup:leliinr-ts whicha t:o the coulc.iri te syst re oc ie _-,peat unit into t:-at of th,, -\scent unit so that: -5 =rc O, C -r as requir:d. The mrltiplication table for thi'.s suT group of the s-Ia. jroap is vel-. sipele. It is:

117 TABLE XIII C2v STAR LINE GROUP MULTIPLICATION TABLE T Ov T ZT Uv There are two irreducible representations, one of which is symmetric to the glide operation, av, and one of which is anti-symmetric to Fav. These are respectively labelled the A and B species. The character table for the fqc0 star group is given in Table XIV. TABLE XIV C2v STAR LINE GROUP CHARACTER TABLE T -v A 1 eiqc/2 B 1 -eiqc/2 Note that the above species notation differs from that of Opaskar, where the A species in his work is symmetric to the C2 operation and the B species is anti-symmetric to C2. Each irreducible representation appears 18 times in the reducible representation since: n(_) = X (Y)(R) X(R) R and: n(A) = 1/2(1 x 36 + eiqc/2 x O = 18 n(B) = 1/2(1 x 36 - eiqc/2 x O) = 18.

118 As a. consequence, there are 18 cartesian syrnzetry coordinates for each species. D. CARTESIAN SYMMETRY COORDINATES The 10 components of the cartesian symmetry coordinate vector: S(7) = U( ) X = U()() U (_) for the yth irreducible representation, are obtained from: Si -= ) X (R) 5i R ThQ results are stown in Eq. (6,4) for the A species, where q = l/(2Nc)1'2 2N-1 s1(A) = inbc C1 S -= e c AX n=O S (A) = Z (.l)n einEc *n! n S(A) - inc cZl n3n n S (A) l= () e ein in AxC s(A) nL S4 7- n n S (A)n ), ( _)n inc AH5 n S ) e czn (A) en 2 in. H3

S10(A)= ~ ) einbc AXC2 n11 Sl )= (-)n =e inc AyCC2 ~10 =1 e cn n S1(A) = ()in in AXH C2 S12(A) = 1 e 1l6 cZC 1n~n n (A) inb 16 = Cij n n s17(A) = 1 (1)l einc yH2 n3 n n S (A) =rl ein ZH2 (6 4)c n for (A) = C/ AZ For the B species, each term in the above coordinates is multiplied by (1l)n = einn. Unlike the ca.se for polyethylene, for which rea.l symmetry coordinates cou.ld be written57 due to the combined presence of the C2 symmetry operation and an opera~tion which takes +q into -q,, this is not possible for polyvinylchloride. The line group for PVC does not include the C2 operation. This is evident from the fact that the molecule has a front a~nd a back, which are defined by the positions of the ch=lorine atoms.. I 9 4 the —.-.-1 - -- -- - - - --- L- -- 1 - I I I fac t ha the moecl _-I a front an a b wh are def ne b y th

120 E. PHASE DEPENDENT DYNAMICAL MATRICES The A and B species symmetry coordinates make it possible to block diagonalize the dynamical matrix C to obtain Cs. There are N/2 (18 x 18) A species blocks for the N/2 values of 5c where 0 < b- < r/2 and there are N/2 (18 x 18) B species blocks for the N/2 values of 5c where c/2 < 6c < it. The N(18 x 18)A and B species blocks for -T _< 6 < 0 are identical to the N blocks for 0 < 5c K< T. However, in this case, all 18 x 18 phase dependent dynamical submatrices are complex valued. To obtain real matrices in this case, the complex dynamical matrix equation, CFcs = FcSA, is first rewritten in row eigenvector form since the computer programs used for these calculations supply a. row eigenvector for each branch j rather than a, column eigenvector. It is then partitioned into real and imaginary submatrices. The complex matrix equation is then: rSS = As. (6 5) The jth row is: U)S saS.2 ~s (6.6) which in real form is: CR IC1 LcRI =-I 2 - ~s(6.7) CI ICR'I /% where CR = C and -CI = CI. -R =R =I =I Therefore the real dynamical matrix is a symmetric matrix. From Eq. (6.7), NS Cs - 7s Cs = i2 s

121 and: ^S ^S "'S 2 2%OS _ _ = ( (6.8) zs cR + zR C = (6. 8) which added together becomes: Ov ^ s /f Ov eV -s ~IV S+ (z+iZ)QR + i(ZR+izI )cI sJ= z1 iz ) or -eV /V Al Ad ( 4+i S )(CS+iC5) = ( 4+is) (6.9) 4 R -I =R =I'' This agrees with Eq. (6.6), as required. The preceding arguments, in terms of CS, are extended to obtain the real (36 x 36) submatrix A(A) ( c). For a, diagonal block, Eq. (3.42) leads to the =0,0 complex matrix: AlA)( )= UAt (A -) A U(A) (65c. 2~=0, O6'=0,7 =0, Y-K since: tl (W (c))7+U,o =0 6 The complex transformation matrix is: cos ~6c + i sin ~5c 0 0o o -18 0 0~(.10)'"-(5 O(6.10)

122 so tihatJ elements in a, given A0 matrix are multiplied by a factor: 2N-1 -2N- ) [cos U6 + i sin C65c][cos (c+y) - i sin(a+y)E] (6.11) a=O Q-=0 which',_: 2N1 - r: 2 2N-1 1 )6c ~ - [cos 73 - i sin 761 (6.13'5? [cos C5 cos (0 + )0 + (co 6 sin )(+7)5] + iAsin I~c cos6 (+issin+6)] ~2~~N: 2 t6=0 (6.14) 22N -1 (),[cos ys - i sin ],(6 o15 a=O whnic.. s A + [A01 x 2N 0 [ T cos yc - i [sin yc O] si (6.17) Io.en th..xpanded re1 matrix is T:~...en th,,.~..xpa~nded r~.=l ma~trix is'

123 _ A+LA0,+ ]Cos 6 -A -A ]sin bc =0_O Y [ 0l,0,1 A(A) ( ) = 0L'A0,+A —,E]c~s S =oO [Ao-(A)(]sin )c AO+o[Ao, +Ao ]cos 6c (6.18) 0,0 Cn -y Ab The real dynamical submatrix Cs o(6c) is obtained finally from Eq. (3.41) where the intermolecular A' (bc) matrix is a, zero matrix. F. DISPERSION CURVES AND POLARIZATION VECTORS The frequency solutions for the isolated syndiotactic polyvinylchloride chain, obtained in this study, are displayed in Fig. 6. 4. (See also Ref. 45. ) Due to the (36 x 36) expanded real form of the dynamical matrix equation, 18 degenerate pairs of frequency solutions are obtained at each 5c value. The squares of the real and imaginary components of the complex eigenvectors, as indicated in Eq. (6.7), are combined to yield the squared polarization vectors, which are given in the Appendix as a, function of 6c. Note that these vector terms are squared values unlike the case for deuterated crystalline polyethylene where the polarization vectors themselves are shown. G. ONE-PHONON FREQUENCY FUNCTIONS The directional amplitude weighted frequency functions for polyvinylchloride are given in Figs, 6.5 to 6.12. The labelling of the atoms, Hi, H2, C~, H5, C1, C2, matches that shown in Figs. 6.1 and 6.2. These spectra are also given the quadratic frequency dependence below.006 eV, in order to calculate the Debye-Waller coefficients.

70 600 550 - V1 50 135o 3oo'(- =. rdn /r ) 250 V1 200 150 - v16 100 ok 20 30 40_L__50 1 1 __v18 0 10 20 30 4 0 50 6 70 80 90go 100 110 10 1 130 140 150 160 170 180 PHASE ANGLE (Sc) Fig. 6.4. Single-chain polyvinylchloride dispersion relations (FT =.06 mdyne - I/rad2).

1500 0?~0 14oo - 1300 o v6 1250 1200 - I8zx I8 1 I I I I io~V 1010 9 9otV9 9000 J 80 1 I 20 0 40 50 60 70 80 90 100 110 120 1I0 140 150 160 170 I 180 PHASE ANGLE (5c) Fig. 6.4. (Concluded).

V V ~~~~~~~~~~~~V2 100 Vil (o) 10 1 o) 0 V1 (ir/2) V16(0) O (i 6V1 rI 17 V8 (oV 0.020.040.060.00.10.120 4 I~. 50.200 V.4 FFIONON ENMEGY (eV) Fig. 6.5. Single-chain polyvinyichioride transverse (H1 or H2) hydrogen frequency function.

V5(O) V12 vl2~~~~~~~~~~~~ v15(_o) v1((N2)2 0HoBo~ ~o~ (ev) Fig. 6.6. Single-chain polyvinylchloride longitudinal (H1 or H2) hydrogen frequency function. 15 ~ ~ ~ ~ ~ 1 H 1 5I52 0.020.040 o06o.080.100.120.140. 0.1 0.200.340.360.380.0 PRONON EIfRGY (eV) Fig. 6.6. Single-chain polyvinyichioride longitudinal (H1 or H2) hydrogen f requency function.

LOO V~~~~~~V V V~~~~~~~~~~ v5(o) lo, — Vl: () o10 v (o) vvl (o) v~ Y15(n[) t2 1 40(7) H, (,r (() V1,~~~~~~~~~~~~~~~~~~~~~~~~~~V a Y~~~~~~15 (ir/ IV16 (0) LFll 7 1 1 I 1 I r II I I lII I- 0.10 llIrJII r I I - __ _I I I- -I /1v.I.IIII o v17 (,/6) 0 v(0) (0) 0.1 0..[020.040.060.,o 0.100.120.140.160.180.200.340.360.380.4 PHON ENERGY (eV) Fig. 6.7. Polyvinylchloride hydrogen (H1 or H2) frequency function for unoriented single-chains.

225.3 V1 10( v7(ir v6 i0) V9 - V14(o) V12 -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ro 1i4 (0vo) PI I B~~~~~~~~~~~~~~~~~~~~ V10(,C/2) U I 1~ V 160) V15(i/2) -15(" V) l111 H~~~~~~~~~~~~~~~~~~~~ V~~~~~ ~~~ ll~ 0 )1i ('/) ~16 Of v4 0.020 *4Q.060.080.100.120.140.160.1.200.3o0.30 PHONON ENERGY (eV) Fig. 6.8. Single-chain polyvinylchloride transverse (H3) hydrogen frequency function.

200 17(O) (5(0) v VV. 10~~~~~~~~~~~1 10 V15 () V4 115(7) 114 (0) v (I( ~~~~~~~~~~~~ Q PU.04 2b 080.200.120.140.60.m.200.34.3b PHON0N ENERGY (eV) Fig. 6.9. Single-chain polyvinyichioride longitudinal (H5) hydrogen frequency function.

10( ~ v () -'~14~~~~~~~~~~~~~~~ v(o) lo-vv(0~~~~~~~~~~~~~~~~~~~~~~~~n,,._., 6(~~) 15 (_/2'-~(o V, (~ V10 LTv. (sr) v (o) v o V~~~~~~vO ~v~o 6rvl o il) v...(0) a~~~~~~~~~~~~~~~~~~~~~~~~ 0~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~.0 0..020 WL O 0o.'.120.1o0.200. 0.3 HowMm Xmar (ev) Fig. 6.10. Polyvinylchloride carbon frequency function for unoriented single-chains.

10C v16 0) F(C V12 V1 (o) 10 vz (2),-~~~~,~ w v15(6/2) V () L~~ ~~~~~~~~~~~~~~~~~~ o r) L~~~~~~~~~~~~~~ pq~~~~~~~~~~~~~~~~~~~~~~~V %~~~~~~~~~~ 01 N tV I I I I I I 0.020.0. 0.80.100.120.140.10.180.200.34o.360.380. PHONON ENMGY (eV) Fig. 6.11. Polyvinyichloride chlorine frequency function for unoriented single-chains.

lODO 112 v8 V16(0) V7(Ir) v16(o)' (6I()'lY ) v./,(, v10o(, /2) V 1 (/6 1) v5 x v11(o) ) 10 iloo V715 (x) Ll~o ) 15kC/-"'I- YVL O I II U VIt5 (AI 1 p I I _________________________'1I 0.020.040.060.080.100.120.10 O.i6o.ieo.200 V. 36.380.)IW -P0oxon VIO( (.v) Fig. 6.12. Polyvinyichioride total frequency distribution, G(E), for unoriented single~-chains. _j~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~,, 0~~~~~~~~~~~~~~~~~~~ o0 %~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~a 0 00 o~o.6 00.0 10io.6 S 20PH'NXN T(V Fig. 6.12. Polyvinylchloride total frequency distribution, G~~~~~~~~~~~~~~~~~~~~e-), for.. unoriented single-chains. ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~~~~~~~~~~~~~~~~~~.

134 H. DIRECTIONAL DEBYE-WALLER COEFFICIENTS The values of the hydrogen atom Debye-Waller coefficients for polyvinylchloride are obtained from Eqs. (2.45) and (2.25) using the frequency functions in Figs. 6.5 to 6.9. The results are shown in Table XV. TABLE XV POLYVINYLCHLORIDE DIRECTIONAL DEBYE-WALLER COEFFICIENTS Transverse Longitudinal 2W or H2) 05A2 2W or H2). o2 2T/ r =.022A2 2WL/2 =.022A2 Approximately 50% of the values for the transverse coefficients and 10% of the values for the longitudinal coefficients are contributed by modes below.006 eV.

CHAPTER 7 POLYTET RAF LUOROEI HYLLE-L As mentioned in chapter I, the modest goal.' tfor this mnaterial isr to simply obtain the dispersion curveso This is only one scttep e>uerld the. results of Krimm and Liang47 in that the same planar approxim.artion is made for the true helical structure and the same diagonal force constants aire utilized~ However, it is felt that these approximate results might be uset'ul to experimentalists attempting to measure the actual dispersion curve -rru ncis by coherent neutron scattering techniques~ A more realistic calculation will be.- required when experimental data. become availableo In this first approximation, since intermolecular forces are relatively weak, a, single chain calculation is performed. The further assumption that the helical configuration may be replaced by a, planar zig-zag structure is also made. Since the rotation angle about the chain axis between any two adjacent CF2 chemical repeat units is 14Ac/13 and it may be assumed also that the intermolecular forces are not long range forces, this is a, reasonable first approach for obtaining the dispersion curveso For the more complete details, one should consult Refo 47. Only the essential points are reviewed here. A. MOLECULAR STRUCTURE AND COORDINATES The unit cell for polytetraf-luoroethylene has been reported by Bunn and Howells8 to be pseudo-hexagonal below 190C and hexagonal above this temperature. There is one helical chain per unit cell, which contains 13 CF2 groups below 135

136 190C and 15 CF2 groups above this temperature. In the planar approximation, the problem becomes identical to that for single-chain polyethylene, except for the magnitudes of the molecular dimensions, atomic masses, and force constants. In this case, the chain dimensions are those given in Table XVI. TABLE XVI PLANAR TEFLON REPEAT UNIT DIMENSIONS c = 2.51l rcF = 1.35A rcc = i 1.54A C-C c-c-c aF F = 1090 28' F-c-F SF-c-c = 1090 28' The internal intramolecular coordinates used here match those for singlechain polyethylene given in chapter 4,. as do the repeat unit cartesian coordinate systems. B. INTRAMOLECULAR FORCE CONSTANTS The force constants which are used in this calculation are the diagonal terms in F_0. (See Eq. (4.3).) These are given in units of 105 dynes/cm or millidynes /A.47 However the B matrix elements used in this reportl8 require bending and torsion force constants in units of millidyne -J/rad.2 Therefore, the force constants used here are those given in Eq. (7.1), which follows.

157 Ar Ar Ar AO A A UC1 A2 a3 nA4 AT CC C-F C-F CCC FCF CCF CCF CCF CCF Ar cc 5346 C-F nrCF 5 37 Arc -F ArcF |537 AGCCC 685 A FCF 1.804 F = XCZ 1.295 =0,0 CCF ~CF SCF 1 1.295,_ACa 4 F 1.295 Ar L0.062 (71) where these values are obtained from the published values as shown below. f = 3.46 m.illidyne/A fF -= 5-37 millidyne/A cCCc x r2 0.289 x (1.54)2 = 0.685 millidyne -A FCF x rC2F = 5,37 x (1.35)2 804 millidyne -A ~CCF x rCF x rCC = 0.623 5 x 1 x 54 = 1,295 millidyne - x r2 o0.026 x (1,54) 0,062 mi.llidyne -A It is interesting to observe tha.t the torsion force consta.nt which is used here is nearly identical to that which was finally used for the crysta~l ]line polyethylene torsion force constant (o0,06) in order to match experimental results for the polyethylene torsion frequency,

138 The cartesian force constant matrices, which are not diagonal, are then obtained by using Eq. (3.48). Since only F O is nonzero: A = (B) F Bs (7.2) =O,m = 0,0 =0, 0 =O,m C. DISPERSION CURVES The frequency solutions to the dynamical matrix equation for polytetrafluoroethylene are displayed in Fig. 7.1.

1800 1700 - 16oo 1 v7 (T) 15oo 1500 F I _ )-v4(T 1400oo - 1300 1200 - v4(T) 1100 -' 1000 goo 80o- v8 (L 700 - ~~600 o(v:s~v(T) ( ) 6oo vT) v5(T) v6 (T) 500 400 o5 —v (L) (L) 200 V L —--- 100 v (T) v (T) 0.2t ( ) 3Ix.52T.6.7t.8 5.9 n PHASE ANGLE (c ) Fig. 7.1. Single-chain Teflon dispersion curves for planar approximation.

CHAPTER 8 CONCLUSIONS AND DISCUSSION In this work, the directional frequency functions for normal and deuterated polyethylene, and polyviny.lchloride have been calculated and used to obtain the directional Debye-Wa.ller coefficients. For a third polymer, polytetrafluoroethylene (Teflon), the dispersion curves have been obtained using a, planar approximation for the true helical molecular structure. In these analyses, the incoherent approximation has been used for the coherent scattering cross-sections of deuterated polyethylene and polyvinylchloride. To obtain the dynamical matrix equations for the frequencies and ca.rtesian polarization vectors, force constant matrices in internal coordinates were first transformed to a cartesian coordinate system and then to a. system of cartesian symmetry coordinates. The polyethylene calculations were first performed for an isolated molecule and then extended to a lattice of such molecules, where intermolecular forces and space group symmetry were considered. A. POLYETHYLTENE The low frequency dispersion curves shown in Figs. 5.4 and 5.5 for crystalline normal and deuterated polyethylene both indicate a, crossover between the v5 and v9 branches between 6 = 0 and 6 l =.it. c c The finite number of frequency values in this 6c region are connected in this way so that there are no discontinuities in the corresponding eigenvector 140

solutions when plotted as functions of 5c. This polarization vector continuity can be seen in Figs. 5.6 and 5.7, where the squares of the vector magnitudes are plotted for the transverse and longitudinal directions. The requirement that the flat dispersion curve branch near 150 cm for 5c <.la be labelled v9 is particularly evident in Fig. 5.7 where the carbon motions are shown. This same requirement is seen in the v5 and v9b curves near 5c =.Ix, in Fig. 5.6. Fig. 5.8, for the transverse one-phonon frequency function of crystalline polyethylene shows great similarity with the corresponding single-chain function in Fig. 4.8. Although the transverse single-chain calculation results in an overestimation of the G-function values below.010 eV, the main features of the spectrum are predicted as well in the single-chain anaylsis as in the analysis for the crystalline lattice. This is not surprising since the intermolecular forces primarily affect the lower vibrational frequencies where v < 200 cm-1. Therefore the transverse G-function obtained from the single-chain analysis is quite satisfactory, except for use in calculations of the Debye-Waller coefficient, where the low frequency contributions are heavily weighted by the cotgh (E/2kT) term as shown in Eq. (2.27). At higher temperatures, this qualification is no longer necessary since the cotgh (c/2kT) term in the DebyeWaller integrand gives increased weight to the higher frequency modes. The same comments apply in comparing the single-chain longitudinal G-function in Fig. 4.7 with the crystalline longitudinal function in Fig. 5.9. At liquid nitrogen temperature approximately 10% of the longitudinal and transverse Debye-Waller coefficients is contributed by modes below.006 eV. At this temperature, modes below.080 eV provide approximately 40% of the contribution

.142 to 2WL and approximately 60% of the contribution to 2WT. The two-phonon G-function values shown in Fig. 5.12 and 5.13 for crystalline polyethylene are not greatly changed by increasing the temperature from 2~K to 1020K. However, greater increases in temperature produce a. relatively greater increase in the calculated two-phonon contribution. This same temperature relationship is also seen in the Debye-Waller coefficients in Fig. 5.14, where there is little difference between the 0K and 100'K coefficients but there is a greater change withtemperature above 1000K. Since polyethylene becomes increasingly amorphous above the glass transition temperature near 4000K, the results in Fig. 5.14, 5.15, and 5.16 for temperatures above 4000K, are expected to be only approximately correct. For example, the transverse coefficient in Fig. 5..14 is probably less than the true value in the high temperature region where "free" rotations of molecular segments occur. From Fig. 5.16, the ratio between the elastic peak magnitudes for the longitudinal and transverse orientations is 1.16 at 1000K and 1.64 at room temperature, for a neutron energy of E = 030 eV, and a scattering angle of Qs = t/2. The crystalline polyethylene, Debye-Waller factor weighted, one- and twophonon frequency spectra are shown in Figs. 5.17 and 5.18 for the various temperatures from 20K to 3020K. Due to the decrease of the Debye-Waller factors with temperature as seen in Fig. 5.15, the one-phonon peak at.022 eV is depressed with increasing temperature. This Debye-Waller temperature effect ot course also applies to the two-phonon contribution near.044 eV, but this two-phonon term is itself temperature dependent, increasing with temperature as seen in Eq. (2.59) where B = l/kt. Therefore, the amplitudes of the one- and two-phonon

143 peaks become more nearly alike at the higher temperatures. This relationship can also be observed by comparing the room temperature neutron scattering measurements of Donovan to the nitrogen temperature measurements of Myers5'12 for polycrystalline polyethylene. The calculated normal polyethylene directional frequency functions are compared to the experimental frequency functions in Figs. 8.1, 8.2, and 8.3 which follow. These functions show qualitative agreement only. Several features in the above G-functions do show reasonable similarity however. From the data, note: (a) the peak near.024 eV, and (b) the shoulder at about.017 eV, (c) the broad maximum near.040 eV, and (d) the very broad peak near.065 eV in Fig. 8.2. These events correlate well with known features in the calculations. Relative to the above data these are, respectively: (a) the (vg) maximum frequency cutoff [the (v5) intercept, v"(0), occurs very near this cutoff and would not be experimentally resolved], (b) one of the intercepts at 0 or n phase angle for the crystalline mode vb5 most probably vb(o) from the relative response of longitudinal and transverse magnitudes, 5 (c) the broad two-phonon maximum near.044 eV, which correlates approximately with the experimental choice of.040 eV, and (d) the frequency limits for both crystalline branches of vi, which occur indistinguishably close together a.t.068 eV in the longitudinal data. With regard to (d), no attempt has been made to adjust the internal coordinate intramolecular force constants28 to effect a better match with the experimental choice of.~65 eV. In addition to these events, the results give some evidence to confirm the calculated discontinuities near.013 eV, associated with vbw(), and near.009 eV, associated with va( ).

144 45,40 cr 35 z c 30 0 25 I20 w 15 > o5 5.010.020.030.040.050.060.070 0.010.020.030 4.050.060.070 PHONON ENERGY (EV) Fig. 8.1. Frequency function, G(E), for unoriented normal polyethylene (experimental points from Ref. 5 for Ef =.030 eV, T = 1000K, Qs = 90~; G(E) = 1/3 GL(E) + 2/3 GT(E)).

145 45 >-40 z35 LL c 30 0 z 25 0 E 20 LUl J 10woe" * ~* *~I%.. 0.010.020.030.040.050.060.070 PHONON ENERGY (EV) Fig. 8.2. Longitudinal frequency function GL(E), for oriented normal polyethylene (experimental points from Ref. 5; Ef =.030 eV, T = 1000K, Gs = 90~).

146 45 >-40 z35 LJ 7z25LU >15 1.0 I 034)( 7 < 10 kC 0.010.020.030.040.050.060.070 PHONON ENERGY (EV) Fig. 8.3. Transverse frequency function, GT(E), for oriented normal polyethylene (experimental points from Ref. 5; Ef =.030 eV, T = 1000K, Gs = 90~).

147 It may be said that the agreement between the calculated and experimental results in Figs. 8ol to 8~3 is as good as should be expected at this stage of refinement. The theoretical analysis is of course based on an idealized model for polyethylene, ioe., a perfect, harmonic crystal. The forces, partieularly the torsional forces, are not harmonic and the carbon chains do exhibit f'oldino, 49 Some work has been done on the latter effect, but little has been done on anharmonic effects and these should be particularly significant for the torsionai modes below Q025 eVo Finally, the phonon wave vector has'been restricted in the calculation to lie along the chain axiso This Cp.roximation should be evaluated o Bo P')LYVINYICHLORIDE In Fig. 6o4, the dispersion curves for polyviny'lehloride are shown~ Except for the addition of the two lower:frequency torsion b0ra~nches v17 and v18 and the crossover shown for the v13 and V14 branches near 5c = 40~, these results match Opaskar's quite welLo45 An interesting point is that the introduction of the two torsion force constants ot' 006 miJynJ -A/rad2 which were not included in Opaskar' s analysis, causes thth vl.. and v16 branches to now have nonzero frequencies of 91 cm'1 and 41 cm~l, respcttively, at 06 = The v17(0) and v17(n) frequencies should be exactly zero but are not due to a slight discrepancy in obtaining the carte-sian coordirnate force constants, when double-precision matrix multiplication methods glre not utilized in the interest of saving computer time~ This discrepancy, which is not important, results in a small nonzero force on an atom and a, small nonzero frequency for a pure translation or rotation of the entire molecule~ The comparison of the

calculated polyvinylchloride frlequency functiorns with measured values iiust v.iit the results of experiments which are row in pirogress. C POLYTTERAFF IJOROEThHYLJE Each branch of the Teflon dispersion curves in Fig. 7. 1 is labeled according to the direction of' motior of' the modes rela.tive to thkG cLa.n exis. For example, the branches fu; 0 6K, <L t/2 include three longitudinelly polarized a.nd 6 transvers ely Qpoolr ariz-d branches. The branch labels rx'i''er d to, for identification parposes ornly, are those of the C(14k/13) hallf.cal caiJer. modes described by Lisan and Y.ri=ur.47 A more realistic calculation, in which the a.etria. bhelical strrc-tre ad a more complete force field a.re irncluded,.is certainly desirable bt-ore an aLttenipt is rma.de to calculate the G-functiol-.

REFERENCES 1. H, R. Danner, GT. JT Sa.fford, H, Boutin and M. Berger, J. Chemo Physo 4L, 14117 (i. 964) 2. JO So King and JT, L. Donovan, Bull. Am, Phys. Soc. 9, 623 (1964). 3. H. Boutin, H. Prask, S. F. Trevino and H. R, Danner, Proceedings of the I.AEA Conference on the Inelastic Scattering of Neutrons in Solids and Liquids, Bombay, India. (.1964). 4, W. R. Myers TJ. L. Donovan and 1, SO King, J. Chemo Phys. 42, 4299 (1965). 5. W. Myers, Go C. Summerfield and J. So King, J. Chemo. Phys. 44, 184 (1966). 6. S. Trevino, J. Chern Phys. 45, 757 (1966)O 7. J. E. Lynch, G. C. Summerfield, L. A. Feldkamp and J. S. King, J. Chemo Phys. (to'be published). 8. W. R. Myers and PF D. Ra.ndo.lph, N. B. S. Symposium on Molecular Dynamics and Struc.ture of Solids, (-967) (to be pub.lished)Q 9. I, A. F.eJ.dkamp and J, So King, No B. S. Symposium on Molecular Dynamics a:nd Stru.,ture of. Solids, (1967) (to be published)..LC, W. E. M.o're;, F. -, Biszchoff,., To Eschi -.f Jo Kirouac, C. Lajeunesse, S. N.:r.hit and M I, Yea.ter, yN. B. S. Symposium on Molecular Dynamics and. otruc ture of Solids) (9') (to be published). i..o, J, L. De(-noDvan, Thesis, University of Michigan, (1964)o 12., W. Ro Myers, Thesis, University Mfi Michiga.n, (1965). ]1. 0.' Sur.r m erfieel. d,. Chem. Phys 45, 1079 (1965). ]h.,o (. Weinreich, "ISo.lidLd, Elerr].entary Theory for Advanced Students" Wil.ey, ( -1965)O 1.5 0o r C, S um.rr.e r.Field,; J. M. Ca:rpenter and N, A. Lurie, "lntroduction to the The3ry ohf S.low-Neutron Sc....tt ering", (to be published). 1.6 A. SA joandde!r, A:rk. F'ys. 1.4, 35.5 (1 958), - 70 E, B. Wil.son, 6,T Chem, Phys. 7, 1C47 (...959); 9, 76 (194.1)

150 1.8. E. B. Wilson, J. C. Decius and P. C. Cross, "Molecular Vibrations", McGraw Hill, New York, (1955). 19. T. Shimanouchi, M. Tsuboi and T. Miyazawa, J. Chem. Phys. 355, 1597 (1961). 20. T. Miyazawa, J. Chem. Phys. 29, 2Li6 (1958). 21. L. Boukaert, R. Smoluchowski and E. Wigner, Phys. Rev. 50, 58 (1936). 22. C. W. Bunn, Trans. Faraday Soc. 35, 482 (1939). 23. P. W. Teare, Acta Cryst. 12, 294 (1959). 24. M. Tasumi, T. Shimanouchi and T. Miyazawa, J_. Mol. Spectroscopy 9, 261 (1962). 25. J. Overend and J. R. Scherer, J. Chem. Phys. 32, 1289 (1960). 26. T. Miyazawa and K. Fukushima, J. Mol. Spectroscopy 15, 308 (1965). 27. M. Tasumi, T. Shimanouchi and T. Miyazawa, J. Mol. Spectroscopy 11, 422 (1965). 28. TJ. H. Schachtschneider and R. G. Snyder, Spectrochim, Acta 19, 117 (1963). 29. M. C. Tobin, J. Chem. Phys. 23, 891 (1955). 30. M. C. Tobin, J. Mol. Spectroscopy 4, 349 (1960). 31. R. Zbinden, "Infrared Spectroscopy of High Polymers", Academic Press, New York, (1964). (pg. 34). 32. E. B. Wilson, J. C. Decius and P. C. Cross, "Molecular Vibrations", McGraw Hill, New York, (1955). (pg. 322). 33. H. Winston and R. S. Halford, J. Chem. Phys. 17, 607 (1949). 34. R. Zbinden, "Infrared Spectroscopy of High Polymers", Academic Press, New York, (1964). (pgs. 53, 57). 35 9 J. R. Nielsen and L. H. Berryman, J. Chem. Phys. 17, 659 (1949). 36. Po W. Higgs, Proco Roy. Soc. (London) A220, 472 (1953). 37. T. Miyazawa, Y. Ideguchi and K. Fukushima, J. Chem. Phys. 38, 2709 (1963).

151 38~ Ho Tadokoro, J Chemrr Physo 339j 1558 (1960). 39. M. Tasumi and T. Shimanouchi, J. Chem. Phys. 43, 1245 (1965). 40, M. Tasumi and S. Krimm, J. Chemo Phys. 46, 755 (1967). 41o P. R, Swan, J. Polymer Sci. 56, 403 (1962). 42, C. Y. Liang, S. Krirnm and G. B. B, M. Sutherland, JT. Chem. Physo 25, 543 (1956). 43. P. Wheatley, "Mclecular Structure", Oxford at Clarendon Press, (.1959) (pgo 17)o 44. S. Krimm and C. Y. Liang, J. Polymer Sci. 22, 95 (1956). 4:5o Co Opaskar, Thesis) University of Michigan, (1967)o 46. Go Natta and P. Corradinr, JO Polymer Sci. 20, 251 (1956) o 4-7. C. Y. Liang and SO Krimrm Jo Chema Phys. 25, 563 (1956), 48. C. Wo Bunn and E. Ro Howells, Nature 74, 549 (1954)O 49. G. Jannink and GO C. Surrmerfield, J. App.o PhysO 37, 3953 (1966). 50. B. Wunderlich, J. Chem. Phys. 37, 1207 (1962).

APPENDIX Page I. POLYETHYLENE Intramolecular Internal Coordinate to Cartesian Coordinate Transformation Submatrices, Bn m for Chain-Oriented Coordinate Systems. 153 The row and column labels are given in Eq. (4.1), page 39 of the text. Internal Coordinate Intramolecular Force Constant Submatrices, Fnm 154 The row and column labels are given in Eq. (4.3), page 41 of the text. Intramolecular Internal Coordinate to Cartesian Coordinate Transformation Submatrices, B, for Crystal-Oriented Coordinate Systems. 155 iDeuterated Crystalline Polyethylene Frequency and Polarization Vector Solutions. 156-166 The phase angle varies from 6c = 0 to E in increments of ~/10. The eigenvector components, for crystal-oriented carte-.;ian coordinate systems, are displayed beneath each frequency solution. The j indices label dispersion curve branches. The format for the eigenvector components matches that used in Eq. (5.2C), page 85 of the text, for the matrix columns. Frequencies are given in units of cm-1. 152

153 BZ SUBMATRIX I = -.000000 -.000000.000000 -.000000 -.000000.000000 -.000000 -.000000.000000 -.000000 -.000()0.000000 -.000000 -.000000.000000 -.000000 -.000000.000000 -.000000 -.000000.000000() -.000000 -.000000.000000 -.000000 -.000000.000000 -.512474.000000 -.379961 -.000000 -.000000.000000 -.000000 -.000000.000000 -.000000 -.000000.000000 -.000000 -.000000.000000 -.000000 -.000000.000000.266172 -.,63082.189920 -.000000 -.000000.000000 -.000000 -.000000.000000.266172.963082.180920 -.000000 -.000000.000000 -.000000 -.000000.000000 -.000000 -.000000.000000 -.()00000 -.000000.000000 -.0(0000 -.000000.000000 -.000000 -.()(0000.000000 -.000000 -.000000.000000 -.000000 -.000000.000000.000000 -.22-3?1 -.000000 -.000000 -.000000.000000 -.000000 -.000000.000000 R3 SUBMATRIX I /.559179.000000 -.829047.000000.000000.000000.000000.000000.000000 -.573572.H1915.000000.573572 -.819155 -.000000.000000.000000.000000 -.573572 -.81G155.000000.000000.000000.000000.573572.819155 -.000000 1.024948.000000 -.000000.000000.000000.000000.000000.000000.000000 1.474699 -.000000.000000 -.737329 -.526598 -.000000 -.737329.526598 -.000000 -.088674 -.826297 -.989270.354846.263215.799350.000000.000000.000000 -.088674.8262~7 -.989270.000000.000000.000000.354846 -.263215.799350 -.088674 -.826297.989270.354846.263215 -.799350.000000.000000.000000 -.088674.826297.989270.000000.000000.000000.354846 -.263215 -.799350.nnoo0.098667.000000 -.231504 -.162099 -.156146.231504 -.162099.156146 B4 SUBMATRIX..559179.000000.829047 -.000000 -.000000.000000 -.000000 -.000000.000000.000000559179.000000.82904700000 -.000000 -.000000.000000 -.000000 -.000000.000000 -.000000 -.000000.000000 -.000000 -.000000.000000 -.000000 -.000000.000000 -.000000 -.000000.000000 -.000000 -.000000.000000 -.000000 -.000000.000000 -.512474.000000.379961 -.000000 -.000000.000000 -.000000 -.000000.000000 -.000000 -.000000.000000 -.000000 -.000000.000000 -.000000 -.000000.000000 -.000000 -.000000.000000 -.000000 -.000000.000000 -.000000 -.000000.000000.000000 -.563000000.000000 -.000000 -.000000.000000 -.000000 -.000000.000000.266172 -.563082 -.189920 -.000000 -.000000.000000 -.000000 -.000000.000000.266172.5638j2 -.189920 -.000000 -.000000.000000 -.000000 -.000000.000000 -.000000 -.098867.000000.231504.162099 -.156146 -.231504.162099.156146 B5SUBMATRIX 5 = o1l2.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.225331.000000.000000.000000.000000.000000.000000.000000

.54 Fi SUBMATRIX " _.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.00000.000000.000000.000000.000000.000000.000000.000000.000000 -.002000 -.003000.000000.000000.000000.000000 000000.000000.000000.000000.o 003000 -.002000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000 F2 SUBMATRIX A - = o,:.064000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.351000.)00000.000000.093000.000000.000000.000000 -.058000 -.058000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.261000.000000.000000 -.058000.000000.002000 -.002000.106000 -.024000.000000.261000.000000.000000 -.058000.000000 -.002000.002000 -.024000.106000.000000 -.004000.000000.000000.000000.000000 -.001000.001000.002000 -.002000.000000 -.004000.000000.000000.000000.000000.001000 -.001000 -.002000.002000.000000.000o 00 o000000.000000.000000.000000.000000.000000.000000.000000.000000 F3 SUBMATRIX = oO 4.427000.000000.000000.351P00.000000 -.004000 -.004000.261000.261000.000000.000000 4.546000.016000.000000.000000.000000.000000.000000.000000.000000.000000.01b000 4.546000.000000.000000.000000.000000.000000.000000.000000.351000.000000.000000.901000.000000 -.124000 -.124000 -.124000 -.124000.000000.o00UO0.000000.000000.000000.550000.000000.000000.000000.000000.000000 -.004000.000000.000000 -.124000.000000.666000 -.016000.023000.000000.000000 -.004000.000000.000000 -.124000.000000 -.016000.666000.000000.023000.000000.261000.000000.000000 -.124000.000000.023000.000000.666000 -.016000.000000.261000.000000.000000 -.124000.000000.000000.023000 -.016000.666000.000000.000000. 00 000 0.000000.000000.000000.000000.000000.000000.000000.107000

155 B2 SUBMATRIX 5 -.000000 -.000000.000000 -.000000 -.000000.000000 -.000000 -.000000.000000 -.000000 -.000000.000000 -.000000 -.000000.000000 -.000000 -.000000.000000 -.000000 -.000000.000000 -.000000 -.000000.000000 -.000000 -.000000.000000 -.362411 -.402488 -.364556 -.000000 -.000000.000000 -.000000 -.000000.000000 -.000000 -.000000.000000 -.000000 -.000000.000000 -.000000 -.000000.000000.602325 -.162247.191043 -.000000 -.000000.000000 -.000000 -.000000.000000 -.224334.582066.191043 -.000000 -.000000.000000 -.000000 -.000000.000000 -.000000 -.000000.000000 -.000000 -.000000.000000 -.000000 -.000000.000000 -.000000 -.000000.000000 -.000000 -.000000.000000 -.000000 -.000000.000000.174424 -.157056.000000 -.000000 -.000000.000000 -.000000 -.000000.000000 B3 SUBMATRIX B1.37471~.416153 -.828496.000000.000000.000000.000000.000000.000000 -.995400.095802.000000.995400 -.095802 -.000000.000000.000000.000000.199343 -.979930.000000.000000.000000.000000 -.199343.979930 -.000000 -.724821 -.804976 -.000000.000000.000000.000000.000000.000000.000000 1.005416 1.116643.000000 -.089600 -.930276 -.000000 -.915816 -.186368 -.000000.559719 -.606396 1.012269.042606.444148.821226.000000.000000.000000 -.661614.493274 1.012274.000000.000000.000000.437281.088792.821231.559719 -.606396 1.012269.042606.444148 -.821226.000000.000000.000000 -.661614.493274 1.012274.000000.000000.000000.437281.088792 -.821231 -.082275.074070 -.000006 -.027815 -.289004 -.157747.284514.057878.157753 B fSUBMATRIX B ~,L.374715.416153.828496 -.000000 -.000000.000000 -.000000 -.000000.000000 -.000000 -.000000.000000 -.000000 -.000000.000000 -.000000 -.000000.000000 -.000000 -.000000.000000 -.000000 -.000000.000000 -.000000 -.000000.000000 -.362411 -.402488.364556 -.000000 -.000000.000000 -.000000 -.000000.000000 -.000000 -.000000.000000 -.000000 -.000000.000000 -.000000 -.000000.000000 -.000000 -.000000.000000 -.000000 -.000000.000000 -.000000 -.000000.000000 -.000000 -.000000.000000 -.000000 -.000000.000000 -.000000 -.000000.000000.602325 -.162247 -.191043 -.000000 -.000000.000000 -.000000 -.000000.000000 -.224334.582066 -.191043 -.000000 -.000000.000000 -.000000 -.000000.000000.082275 -.074070 -.000006.027815.289004.157747 -.284514 -.057878.157753 BS SUBMATRIX B.00000o.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.00000O.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.000000.174424.157056.000000.000000.000000.000000.000000.000000.000000

156 DELC =.000000, FREQ = 997.603951, 1 A4~c4C AC, cc A bt Ac AC 46 L~.547434 -.464094 -.000000.365324.399312 -.000000 -.245331 -.363089 -.000000 DELC =.000000, FREQ = 1094.754761, J = 2.588432.675242.000000.299143 -.083029.000000 -.100996.302018.000000 DELC =.000000, FREQ =.000000, J = 3.000000.000000.865342.000000.000000.354389.000000.000000.354389 DELC =.000000, FREQ = 2132.675598, J = 4 -.308561 -.314755 -.000000.651512 -.063182 -.000000 -.117899.602793 -.000000 DELC =.000000, FREQ = 142.945490, J = 5 -.358459.323641 -.000000 -.099435.616034 -.000000 -.613158.037617 -.000000 DELC =.000000, FREQ = 792.973564, J = 6.000000.000000 -.000009.000000.000000.707118.000000.000000 -.707095 DELC =.000000, FREQ = 1032.205185, J = 7 -.004361.082719 -.000000.055773.664110 -.000000.716069.190343 -.000000 DELC =.000000, FREQ = 2207.879791, J = 8.361028 -.343524.000000 -.582740.095516.000000 -.164300.613145.000000 DELC =.000000, FREQ = 913.943954, J = 9.000000.000000 -.501182.000000.000000.611876.000000.000000.611903 DELC =.000000, FREQ = 999.334702, J = 10.549626 -.470874.000000.360247.356553.000000 -.283746 -.372516.000000 DELC =.000000, FREQ = 1093.102142, J = 11.585201.670058.000000.295865 -.122371.000000 -.141563.293274.000000 DELC =.000000, FREQ = 53.880099, J = 12.000000.000000.868342.000000.000000.348935.000000.000000.352459 DELC =.000000, FREQ = 2128.322968, J = 13 -.310313 -.316462.000000.650785 -.063834.000000 -.119452.601409.000000 DELC =.000000, FREQ = 1047.939346, J = 14.056489.097444 -.000000.095387.686737 -.000000.686889.186523 -.000000 DELC =.000000, FREQ = 789.504189, J = 15.000000.000000.008235.000000.000000.700409.000000.000000 -.1l3694 DELC =.000000, FREQ = 108.936782, J = 16.355900 -.320132.000000.098846 -.610780.000000.622041 -.033058.0(00000 DELC =.000000, FREQ = 2211.227386, J = 17.359584 -.342302.000000 -.583349.095710.000000 -.162566.614528.000000 DELC =.000000, FREQ = 918.757133, J = 18.000000.000000 -.495898.000000.000000.622633.000000.000000.605321

157 DELC.314159, FREO = 1000.781464, J = 1.525744 -.452732 -.015250.352104.387050.203955 -.227042 -.351813 -.166408 DFLC =.314159, FREQ = 1090.275833, J = 2.570988.649540 -.085295.285366 -.126474.154250 -.147575.279170.154461 DELC =.314159, FREQ = 205.822094, J = 3 -.042409 -.019729.844931 -.013765.026602.377692 -.015632 -.007097.374247 DELC =.314159, FREQ = 2130.786255, J = 4 -.305310 -.312310.001223.652260 -.061983 -.009350 -.117374.604986 -.009421 DELC =.314159, FREQ = 145.614998, J = 5 -.361811.328828 -.029504 -.101649.611115.024500 -.609231.040767 -.053631 DELC =.314159, FREQ = 780.580505, J = 6 -.128414.110905 -.002685 -.080187 -.126229.682805.116182.088490 -.679181 DELC =.314159, FREQ = 1032.478943, J = 7.016110.111444 -.070908.064884.652934.061350.705759.202100.095159 DELC =.314159, FREQ = 2206.584595, J = 8.360018 -.341770.000049 -.584403.094815.009813 -.163289.613344 -.010658 DELC =.314159, FREQ = 912.478851, J = 9 -.15900P -.159845 -.522185 -.080351 -.070280.566733 -.056858 -.075536.578764 DFLC =.314159, FREQ = 1001.703423, J = 10.528875 -.461193 -.008791.347054.335776.192419 -.275835 -.364415 -.168563 DELC =.314159, FREQ = 1089.793564, J = 11.562225.639829 -.085343.277498 -.177636.156004 -.195988.266918.157112 DELC =.314159, FREQ = 213.128660, J = 12 -.018400 -.041818.848266 -.007553 -.013503.369461.023728 -.010411.375427 DELC =.314159, FREQ = 2126.495178, J = 13 -.306995 -.313940.001170.651560 -.062600 -.008967 -.118703.603737 -.008733 DELC =.314159, FREQ = 1047.289902, J = 14.090206.128108 -.075875.111310.673686.090795.669598.196665.090380 DFLC =.314159, FREQ = 777.999229, J = 15 -.124111.109664.006209 -.076497 -.124849.678624.110325.085357 -.686370 DELC =.314159, fREQ = 113.779606, J = 16.361171 -.323045 -.024118.101441 -.606735 -.050889.617770 -.035581.026477 DELC =.314159, FREQ = 2209.895599, J = 17.356609 -.340588.000079 -.585011.094953.009767 -.161749.614622 -.011240 DELC =.314159, FREQ = 916.009216, J = 18 -.164387 -.169753 -.516435 -.084795 -.070812.574975 -.050138 -.079751.570771

158 DELC =.628318, FREQ = 1081.374283, J = 1.511316.563308 -.167773.242268 -.247802.280585 -.277712.209407.270718 DELC =.628318, FREQ = 1007.215652, J = 2 -.473447.430948.030922 -.319248 -.348819 -.357959.196197.330019.300068 DELC =.628318, FREQ = 378.183083, J = 3 -.092457 -.088624.771063 -.033820.009130.442911 -.001376 -.028226.436878 DELC =.628318, FREQ = 2125.582458, J = 4 -.296319 -.305594.000431.654200 -.058630 -.016953 -.115913.611039 -.017294 DELC =.628318, FREQ = 152.289717, J = 5 -.376375.342106 -.008983 -.109842.597770.070778 -.597480.049732 -.085820 DELC =.628318, FREQ = 747.463982, J = 6 -.222892.192234 -.008977 -.138747 -.234977.621667.216290.154853 -.617728 DELC =.628318, FREQ = 1034.457535, J = 7.051416.146850 -.221093.078875.602126.187135.646393.206751.250085 DELC =.628318, FREQ = 2202.859772, J = 8.357030 -.336771.000040 -.589172.092786.018628 -.160396.613924 -.020103 DELC =.628318, FREQ = 911.476860, J = 9 -.310802 -.326733 -.572113 -.160010 -.172308.412108 -.153721 -.164219.440010 DELC =.628318, FREQ = 1006.389977, J = 10.480839 -.436514 -.006752.315739.283270.332836 -.260481 -.344916 -.317288 DELC =.628318, FREQ = 1083.659546, J = 11.487902.547941 -.154686.223401 -.317846.262114 -.332551.194127.264331 DELC =.628318, FREQ = 382.383827, J = 12 -.080403 -.101644.774011 -.031521 -.008945.433883.014865 -.032033.439926 I)ELC =.628318, FREQ = 2121.493164, J = 13 -.297793 -.307036.000294.653528 -.059155 -.016171 -. 116593.610196 -.015941 DELC =.628318, FREQ = 1046.570511, J = 14.144386.163375 -.229411.132903.609737.250190.594290.195930.237116 DELC =.628318, FREQ = 746.680359, J = 15 -.220633.191493.001041 -.135470 -.233655.622857.207814.151065 -.622692 OELC =.628318, FREQ = 126.002691, J = 16.375763 -.3 15070 -.011113.108858 -.594460 -.086498.605885 -.044171.068145 DELC =.628318, FREQ = 2206.055817, J = 17.355747.335682.000107 -.589812.092757.018556 -.159425.614871 -.021245 DELC =.628318, FREQ = 912.300591, J = 18 -.313560 -.341024 -.569367 -.162984 -.163681.416687 -.137344 -.169572.431975

159 DELC =.942478, FREQ = 1011.664482, J = 1.408980 -.412384 -.022229.276123.280700.4,9804 -.186758 -.312764 -.425459 DELC =.942478, FREQ = 1076.830292, J = 2.417241.437043 -.272727.184546 -.346828.363180 -.383623.126960.333154 DELC =.942478, FREQ = 913.407898, J = 3.418983.452981.558847.223303.303050 -.102339.279878.244523 -.129037 DELC =.942478, FREQ = 2118.262634, J = 4 -.283564 -.296274 -.003719.656510 -.053753 -.021613 -.113817.619658 -.022512 DELC =.942478, FREQ = 158.728916, J = 5 -.401034.365333 -.005667 -.123883.572470.109958 -.575121.065719 -.124210 DELC =.942478, FREQ = 701.836998, J = 6 -.274451.245707 -.015694 -.171796 -.327058.539698.304147.200099 -.551142 OELC =.942478, FREQ = 1052.548828, J = 7 -.017381.019414 -.478171.039809.512742.303919.527930.120940.347102 DELC =.942478, FREQ = 2197.154938, J = 8.352261 -.329212 -.000030 -.596491.089655.025540 -.155995.614783 -.027323 OELC =.942478, FREQ = 477.377529, J = 9 -.205299 -.208243.619577 -.081125 -.023590.517056 -.018815 -.073134.500423 DELC =.942478, FREQ = 1009.217667, J = 10.435374 -.391677.010468.284699.233184.425645 -.231021 -.311707 -.435654 DELC =.942478, FREQ = 914.384033, J = 11.413509.467411.571855.219270.278276 -.110584.259615.250147 -.125696 DELC =.942478, FREQ = 1080.435974, J = 12 -.391109 -.437370.219055 -.161356.429707 -.305121.438173 -.120165 -.312310 DELC =.942478, FREQ = 2114.550995, J = 13 -.284679 -.297521 -.003979.655753 -.054155 -.020426 -.113492.619482 -.020564 DELC =.942478, FREQ = 1060.477615, J = 14.071926.062007 -.490383.084379.487952.378640.472106.123676.351768 DELC =.942478, FREQ = 702.376694, J = 15 -.280663.243742 -.005685 -.172756 -.326449.545860.293893.195962 -.550121 DELC =.942478, FREQ = 139.740850, J = 16.401273 -.355682 -.008450.122015 -.571435 -.125478.583724 -.059309.104765 DELC =.942478, FREQ = 2200.144836, J = 17.351226 -.328200.000081 -.597272.089360.025477 -.155925.615145 -.028968 DELC =.942478, FREQ = 479.121616, J = 18 -.192680 -.225440.619913 -.078947 -.037631.505327 -.007018 -.080311.508189

160 DELC = 1.256637, FREO = 1010.824532, J = 1.361307 -.369793 -.003258.242428.238180.497213 -.153634 -.278351 -.518552 DLLC = 1.256637, FREQ = 906.547028, J = 2.410872.450128.433451.224831.352077.225747.324058.254403.213272 DELC = 1.256637, FREO = 1114.472656, J = 3 -.008591 -.011714.777132 -.026141 -.204912 -.390550 -.199266 -.048557 -.398232 OELC = 1.256637, FREO = 2110.288300, J = 4 -.269487 -.286488 -.011290.658154 -.048129 -.022977 -.111478.629386 -.024657 DELC = 1.256637, FREQ = 1075.785873, J = 5 -.340767 -.337902.074336 -.124094.556837 -.233525.592302 -.036660 -.178905 DELC = 1.256637, FREQ = 651.822311, J = 6 -.287793.279020 -.016576 -.183995 -.408240.452315.389003.231331 -.478652 )tLC = 1.256637, FREO = 159.950380, J = 7.437441 -.398341.003826.144538 -.532263 -.146914.539270 -.089404.159281 I)LLC = 1.256637, FREQ = 2190.127747, J = 8.346155 -.319968 -.000104 -.605668.085784.029847 -.150595.615657 -.031627 DELC = 1.256637, FREQ = 481.544056, J = 9 -.343247 -.354034.449702 -.140891 -.087274.516878 -.060100 -.132023.488812 i)LLC = 1.256637, FRFQ = 1007.707710, J = 10.396541 -.334971.010464.257201.198085.512403 -.187889 -.268919 -.504850 UELC = 1.256637, FREQ = 910.801186, J = 11.404410.494088.446771.217331.326400.218504.319682.262929.220517 DFLC = 1.256637, FREQ = 1116.404633, J = 12 -.008862 -.000999.766078 -.026069 -.231911 -.394748 -.229094 -.047642 -.384709 D~LC = 1.256637, FREO = 2107.147919, J = 13 -.27010F -.287643 -.011694.657088 -.048415 -.021422 -.109901.630098 -.022262 O)tLC = 1.256637, FREQ = 1079.670853, J = 14 -.307863 -.392251.111791 -.104308.573447 -.207726.574381 -.052884 -.230348 DEIC = 1.256637, FREQ = 652.413956, J = 15 -.304264.274989 -.010206 -.190583 -.408678.457098.376268.226612 -.475910 DELC = 1.256637, FREW = 149.265955, J = 16.438676 -.385828 -.007746.141698 -.534487 -.162094.548505 -.082030.137919 I)ELC = 1.256637, FKEO = 2192.811707, J = 17.345507 -.318952.000064 -.606770.085137.029829 -.151691.615177 -.033668 D)ELC = 1.255637, FREQ = 479.538918, J = 18 -.3 27109 -.373340.447896 -.136821 -.096508.504143 -.050797 -.141191.498293

DELC = 1.570796, FREQ = 413.388908, J = 1.442235.471298 -.339096.182231.147165 -.440132.105756.181015 -.418076 DELC = 1.570796, FREQ = 604.712837, J = 2 -.280513.283069 -.009858 -.184980 -.484659.373538.469705.247026 -.388328 DELC = 1.570796, FREQ = 1195.137024, J = 3.073043.079447.867528.024557 -.161321 -.299148 -.161359.006764 -.305856 DELC = 1.570796, FREQ = 2102.869934, J = 4 -.256159 -.278050 -.020662.658331 -.042473 -.021471 -.109266.639113 -.023965 DELC = 1.570796, FREQ = 1079.756912, J = 5 -.293769 -.291255.130063 -.096451.583047 -.237192.610684 -.012563 -.182631 I)ELC = 1.570796, FREQ = 1004.022598, J = 6.318032 -.318796.000626.210689.201106.554090 -.120443 -.238513 -.577910 DELC = 1.570796, FREQ = 151.342007, J = 7.485321 -.438143.001622.171585 -.474546 -.177270.488038 -.119944.184021 DELC = 1.570796, FREQ = 2182.539734, J = 8.33939~ -.309941 -.000094 -.615973.081617.031144 -.144753.616177 -.032696 DFLC = 1.570796, FREQ = 891.810211, J = 9.341497.373675.339065.189772.321522.425596.289208.215857.421929 DELC = 1.570796, FRFQ = 408.336830, J = 10.426471.484986 -.338002.177454.153274 -.434443.096379.187541 -.424830 DELC = 1.570796, FREQ = 603.906479, J = 11 -.299797.282574 -.009919 -.194503 -.485063.368013.456770.243984 -.391736 UOLC = 1.570796, FREQ = 1195.352020, J = 12.068167.080085.865506.024866 -.164941 -.304135 -.168222.006983 -.301942 DFLC = 1.570796, FREQ = 2100.463379, J = 13 -.256133 -.279284 -.021178.656669 -.042680 -.019660 -.106334.640888 -.021357 DFLL = 1.570796, FREQ = 1080.946915, J = 14 -.2<'511] -.308061.137733 -.080260.601039 -.191591.600488 -.027112 -.223889 DELC = 1.570796, FRED = 1001.051987, J = 15.3'7045 -.286442.002291.222733.156488.577749 -.155544 -.230526 -.558817 DELC = 1.570796, FREQ = 149.893726, J = 16.486830 -.425108 -.007574.167754 -.480629 -.190287.497341 -.112656.163790 I)ELC = 1.570796, FRED = 2184.824951, J = 17.339249 -.308794.000145 -.617623.080549.031195 -.147182.614614 -.034969 DELC = 1.570796, FREQ = 896.601334, J = 18.339336.384064.342161.182196.298270.418638.297784.224041.428933

162 DELC = 1.884955, FREQ = 130.332090, J = 1.542991 -.474792 -.002911.203572 -.401149 -.193592.425912 -.152252.187890 DELC = 1.884955, FREQ = 306.859070, J = 2.492762.554151 -.274847.201469.191011 -.342286.141604.216153 -.336975 DELC = 1.884955, FREQ = 1263.681915, J = 3.093464.102375.903358.043709 -.126408 -.251477 -.129090.029792 -.256994 DELC = 1.884955, FREQ = 2096.657013, J = 4 -.244754 -.271815 -.028624.656867 -.037354 -.017995 -.107470.648232 -.021078 DELC = 1.884955, FREQ = 1084.929001, J = 5 -.263449 -.266278.134131 -.076905.610378 -.204801.629229.001357 -.159213 DELC = 1.884955, FREQ = 992.075516, J = 6.269717 -.265863 -.000963.175656.160266.601818 -.095022 -.197792 -.624253 OELC = 1.884955, FREQ = 566.519226, J = 7 -.262025.259356 -.002867 -.179691 -.554477.296195.538661.247871 -.291560 DELC = 1.884955, FREQ = 2175.155182, J = 8.332740 -.300012.000045 -.626607.077596.029367 -.139018.616032 -.030575 DELC = 1.884955, FREQ = 877.436981, J = 9.268662.290285.299291.151293.271120.537657.236417.170130.532608 OELC = 1.884955, FREQ = 138.881344, J = 10.537779 -.472568 -.005923.196698 -.412275 -.199333.430568 -.150059.176457 DELC = 1.884955, FRED = 300.275230, J = 11.489936.553408 -.274924.200969.192374 -.348234.136606.216187 -.337725 DELC = 1.884955, FREQ = 1263.315582, J = 12.089843.102388.903689.045035 -.125611 -.253504 -.130788.030386 -.254344 DELC = 1.884955, FREQ 2176.979706, J = 13 -.333184.298591 -.000362.629036 -.076078 -.029483.142825 -.613196.032858 DELC = 1.884955, FRED = 1083.168060, J = 14 -.241434 -.281899.135592 -.067321.621771 -.160985.622812 -.010117 -.197007 DFLC = 1.884955, FREQ = 989.985344, J = 15.289691 -.239734 -.003650.183742.116293.621852 -.126857 -.192190 -.609572 DELC = 1.884955, FREQ = 563.150703, J = 16 -.275121.265670 -.008217 -.188638 -.554360.279843.529669.247927 -.300717 DELC = 1.884955, FREQ = 2095.064087, J = 17 -.244110 -.273304 -.029167.654326 -.037532 -.016113 -.103239.651192 -.018543 DELC = 1.884955, FREQ = 880.247253, J = 18.272360.297596.297338.145558.250216.531674.251497.176786.536534

163 DELC = 2.199114, FREQ = 97.789557, J = 1.621135 -.476519 -.016644.243431 -.314527 -.193992.370673 -.170703.156483 DELC = 2.199114, FRED = 190.154552, J = 2.488491.632785 -.223942.198826.235766 -.242368.154705.249599 -.265923 DELC = 2.199114, FREQ = 1307.104034, J = 3.091910..100777.920057.048183 -.102168 -.231688 -.105883.037069 -.236361 DELC = 2.199114, FRED = 2168.662537, J = 4 -.326910.291048 -.000254.636614 -.074103 -.024772.133905 -.615183.025610 DELC = 2.199114, FRED = 1091.041275, J = 5 -.248050 -.256544.124434 -.066000.629724 -.165027.641595.007055 -.131666 DELC = 2.199114, FREQ = 977.074600, J = 6.213689 -.209006 -.004141.136676.118592.639032 -.072614 -.154449 -.663628 DELC = 2.199114, FREQ = 540.046822, J = 7 -.237130.223810.001627 -.170429 -.609166.217587.589791.240128 -.201712 O)ELC = 2.199114, FREQ = 2091.778595, J = 8 -.23511 -.267643 -.031482.654313 -.033183 -.013539 -.106259.656417 -.016732 DELC = 2.199114, FREQ = 865.292030, J = 9.203260.215464.294232.115767.216233.605152.182932.127620.591243 o)ELC = 2.199114, FREQ = 183.439222, J = 10.569304.557767 -.225202.230466.199081 -.277153.192429.220705 -.242427 DELC = 2.199114, FRED = 116.630153, J = 11 -.547635.557157 -.010853 -.209549.352891.165162 -.345735.202924 -.181995 DELC = 2.199114, FREQ = 1306.554947, J = 12.089070.100607.921093.049770 -.099952 -.231758 -.105387.037699 -.234147 I)ELC = 2.199114, FREQ = 2170.019440, J = 13 -.327949.289238 -.000623.639989 -.072153 -.024921.138988 -.610978.027650 DELC = 2.199114, FREQ = 1086.445755, J = 14 -.232036 -.269245.123363 -.062814.634812 -.129678.639017 -.000495 -.162346 OELC = 2.199114, FREQ = 976.446892, J = 15.225642 -.189478 -.008072.141456.080954.650825 -.097642 -.150653 -.656316 DELC = 2.199114, FREQ = 533.863037, J = 16 -.237325.235135 -.007027 -.175872 -.608739.194635.587417.242448 -.212419 DELC = 2. 199114, FREQ = 2090.968231, J = 17 -.234278 -.269507 -.031949.650701 -.033375 -.011819 -.100925.660602 -.014559 DELC = 2.199114, FREQ = 865.126381, J = 18.211045.219940.290516.112308.198548.602629.199731.132368.591673

164 DELC = 2.513273, FREQ = 51.836852, J = 1.781362 -.279541 -.062599.315033 -.165261 -.190026.360609 -.105839.059189 DELC = 2.513273, FREQ = 98.540828, J = 2.283138.787327 -.154736.115720.316009 -.110440.082565.314600 -.211577 DELC = 2.513273, FREQ = 1326.728043, J = 3.074578.081761,.930579.040858 -.077143 -.228888 -.080543.032623 -.232840 DELC = 2.513273, FREQ = 2163.621521, J = 4 -.322420.283894 -.000373.644891 -.071424 -.017870.129874 -.613953.018378 DELC = 2.513273, FREQ = 1097.521652, J = 5 -.243441 -.256253.100431 -.061600.643474 -.118885.650687.007910 -.097675 DELC = 2.513273, FREQ = 854.882103, J = 6.140665.146192.308666.080708.154953.649010.127970.087112.622787 DELC = 2.513273, FREQ = 524.275421, J = 7 -.214891.190279.003646 -.161433 -.645553.141867.621581.230010 -.124133 DELC = 2.513273, FREQ = 2088.171814, J = 8 -.228765 -.264950 -.026959.651638 -.030196 -.008866 -.105632.663195 -.011529 DELC = 2.513273, FREQ = 962.194542, J = 9 -.149090.145768.007713 -.093851 -.077804 -.665906.049986.106951.695869 DELC = 2.513273, FREQ = 107.587630, J = 10.834074.059736 -.122912.335077 -.011940 -.230144.344015.025089 -.035827 DELC = 2.513273, FREQ = 68.362881, J = 11 -.057584.828588 -.114759 -.019643.370543 -.026739 -.077186.327903 -.212308 DELC = 2.513273, FREQ = 1326.155029, J = 12.072412.081474.931830.042249 -.074640 -.227459 -.079163.033094 -.230988 DELC = 2.513273, FREQ = 2164.573151, J = 13 -.323977.281668 -.000718.649212 -.069115 -.018005.135987 -.608469.019916 DELC = 2.513273, FREQ = 1090.584213, J = 14 -.231699 -.266211.098361 -.063388.643838 -.094643.651255.003714 -.118837 DELC = 2.513273, FREQ = 851.733841, J = 15.149392.148389.304654.079201.141827.651551.142472.090015.619307 DELC = 2.513273, FREQ = 515.946922, J = 16 -.198278.204909 -.005908 -.161179 -.643511.118727.626201.233631 -.133304 DELC = 2.513273, FREQ = 2088.004669, J = 17 -.226810 -.267208 -.027283.646952 -.030423 -.007546 -.099478.668496 -.009952 DELC = 2.513273, FREQ = 963.253090, J = 18 -.155048.132748.012133 -.096222 -.050335 -.668302.066942.104560.695957

165 DELC = 2.827432, FREQ =.000000, J = 1.816984 -.261302 -.037362.333611 -.113149 -.103272.340784 -.105379.029324 DELC = 2.827432, FREQ = 76.218102, J = 2.257784.821732 -.083305.104741.319032 -.051642.104072.335546 -.111905 DELC = 2.827432, FREQ = 1331.327957, J = 3.042520.046601.938208.023517 -.043335 -.233832 -.045339.018967 -.237194 DELC = 2.827432, FREQ = 2160.436218, J = 4 -.319600.279278 -.000276.650367 -.069748 -.009350.127295 -.612890.009584 DELC = 2.827432, FREQ = 1162.953796, J = 5 -.244733 -.260225.057999 -.061074.652784 -.063557.657445.006602 -.053388 DELC = 2.827432, FREQ = 951.135406, J = 6.076848 -.075351 -.010687.047846.038392.682393 -.025717 -.054978 -.717771 DELC = 2.827432, FREQ = 516.4D00757, J = 7 -.201375.166953.002987 -.155769 -.665922.070033.637387.221976 -.058392 DELC = 2.827432, FREQ = 2085.886597, J = 8 -.224418 -.263349 -.015508.649712 -.028445 -.004346 -.105411.667790 -.005866 DELC = 2 -'32, FREQ = 846.797813, J = 9.073601.0-.2.328179.042357.082529.676225.067129.045018.639355 DELC = 2.827432, FREQ = 101.816755, J = 10.857676.109425 -.071247.348999.055558 -.122861.324437.039538 -.023157 DELC = 2.827432, FREQ =.000000, J = 11 -.115675.849648 -.058588 -.046196.352246 -.004265 -.055940.345901 -.110803 DELC = 2.827432, FREQ = 1330.801895, J = 12.041267.046350.939431.024338 -.041598 -.231198 -.044192.019204 -.235629 DELC = 2.827432, FREQ = 2161.112091, J = 13 -,.321512.276730 -.000490.655399 -.067201 -.009430.134076 -.606475.010413 DELC = 2.827432, FREQ = 1094.456726, J = 14 -.235317 -.268332.056339 -.065917.650233 -.051290.659960.004578 -.064053 DELC = 2.827432, FREQ = 841.382629, J = 15.079471.076228.324475.041992.075427.683494.075696.046379.632528 DELC = 2.827432, FREQ = 506.881020, J = 16 -.169431.184773 -.003714 -.149758 -.660837.055020.647670.226751 -.063917 DELC = 2.827432, FREQ = 2086.139801, J = 17 -.222021 -.265904 -.015668.644218 -.028706 -.003628 -.098759.673886 -.005041 DELC = 2.827432, FREQ = 953.572533, J = 18 -.079039.068822.015584 -.048710 -.023819 -.677133.034280.053818.723361

166 DELC = 3.141590, FREQ =.000000, J = 1.787953 -.364522 -.000000.323280 -.137407 -.000001.316799 -.150005.000000 DELC = 3.141590, FREQ = 84.951834, J = 2.358371.791752 -.000001.145955.304075 -.000000.156990.326019 -.000001 DELC = 3.141590, FREQ = 1331.270828, J = 3.000000.000000.941330.000000 -.000000 -.237075 -.000000.000000 -.240193 DELC = 3.141590, FREQ = 2159.348022, J = 4 -.318639.277684 -.000000.652282 -.069179 -.000000.126407 -.612474.000000 DELC = 3.141590, FREQ = 1105.162888, J = 5 -.246171 -.262589.000000 -.061415.656335 -.000000.660202.005735 -.000000 DELC = 3.141590, FREQ = 947.031326, J = 6.000001 -.000001 -.011872.000000.000000.688010 -.000000 -.000000 -.725604 DELC = 3.141590, FREQ = 514.065903, J = 7 -.197180.158516.000000 -.153988 -.672547.000001.641906.218899 -.000000 DELC = 3.141590, FREQ = 2085.093597, J = 8 -.222911 -.262799 -.000000.649023 -.027875 -.000000 -.105371.669429 -.000000 DELC = 3.141590, FREQ = 843.547813, J = 9.000001.000001.337278.000000.000001.685885.000001.000000.644830 DELC = 3.141590, FREQ = 110.360845, J = 10.844576.222291 -.000001.345295.117478 -.000001.311651.084393 -.000000 DELC = 3.141590, FREQ =.000000, J = 11 -.231782.836580 -.000000 -.094576.334117.000000 -.083120.344852 -.000001 DELC = 3.141590, FREQ = 1330.777542, J = 12.000000.000000.942489.000000 -.000000 -.233926 -.000000.000000 -.238734 DELC = 3.141590, FREQ = 2159.926178, J = 13 -.320677.275015 -.000000.657579 -.066549 -.000000.133419 -.605717.000000 DELC = 3.141590, FREQ = 1096.112518, J = 14 -.237446 -.270034.000000 -.067276.652780 -.000000.663350.004463 -.000000 OELC = 3.141590, FREQ = 837.262680, J = 15.000001.000001.333805.000000.000001.695134.000001.000000.636682 DELC = 3.141590, FREQ = 504.180912, J = 16 -.158739.178037 -.000000 -.145427 -.665749.000000.654537.224311 -.000000 DELC = 3.141590, FREQ = 2085.492920, J = 17 -.220355 -.265465 -.000000.643227 -.028149 -.000000 -.098553.675815 -.000000 DELC = 3.141590, FREQ = 950.006073, J = 18 -.000001.000001.017016 -.000000 -.000000 -.679756.000000.000000.733241

167 Page II. POLYVINYLCHLORIDE Internal Coordinate to Cartesian Coordinate Transformation Submatrices, Bn, for Chain-Oriented Cartesian Coordinate Systemso 168-169 The row and column labels are given in Eq. (6.1), page 111 of the text. Internal Coordinate Intramolecular Force Constant Submatrices, Fnm 170 The format is given in Eqo (603), page 115 of the text. Frequencies in cm-1 and Squares of Polarization Vector Components for 6c = 0 to 1800 in 200 Incrementso 171-180 The cartesian eigenvector components are displayed beneath each frequency solution where the dispersion curve branches are labelled by the j indices, The format for the cartesian components matches that for the coliLnns-in Eqo (6o1), page 111 of the text.

u ~~ oi 0 o o~ o o o i o: ~: o~ ol o! 0 O(Vc o: o 0 oi. ~ ~~ 0 100 0o C 0 0 01~....j 6 00 01 0 0 0 0 0 0 0 0 0 0 0 0 C.C C) 9 o' o 0' 0 0 C) C) C) C)' C) C) C) C) C 0 0 C) 0 0 0 0 0 0 0 0 0 - 0 0 0 0 0 0 N 0 N 0 0 i 0, C) 0 1 C 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0l 0 0' O C) 0 C) C) C) C....C C C) 0 0 ) 0 0 C 0 0 0 0 - 0 0 oC0 C 0a N o0 N I 4 [ ~ t''.i....''......... C) C) 0 C~~~~~~~l C) C) C), C.~~~~~~~~ C) C) C)~~~~~~ C) C) C)1 0 0 C) 0 0 C) C) C) C) N C) C) U~~~~~~~~~~~~~~~~~~~~Q 0 )C) N C) C)C) C) C))I N 0 1 l ~c C)~ C):C) o o o' o'. o o o o; l r. o C 0'4 o L. - C)u~~ 0u,~~ C)C)C)C)C) C.. l' C) C.) Q C)~ C) C)C) 0; U/ Ui 0!C)C Cl c) 0 a0 C) 0 0 C C) C UC N 0 C if ~;' j g II,i ~,! C-) Ui 0 0 0 C): C-) 0 C C C C C) ) C) ) C N C C) ) C) 0 C; La C-)) C C C-)j C, U Ul L)) t0C)0C)C)O C) ) )C C c) C) C) U) C C) N C) N a) (n C,j I 0 C 0 C) 0 0 01 C)' 0 OD U 0C C C) C 0 C ) N C C) N C-) 0l C)' V )' C? C' 0 C), CCL U 01,j O L,0100010, C) C) C) C) C) C.) C) 0L) C) 0U C) a C)NC) VI, O ~ ~ ~ ~ ~ ~o 0 C), iL lw 4.) U - I Oi-L 01 C) 0 ) C) C) C C) C) C) N C) C) C) C) C) N C) N C)' C 01 0., Cd U C) 0Q C) CL" C) C) C) C) L) ) C)U C) f- C) C) 0 C) N C) N C) 0 C )1 C) VIUL ) C) U U U L) 0D U C) C) Q U U C) ) C)' C-) C) C) C) C) w) C' C 0 C) CU f C 0 C ) C) C) C) 0CD )C C)C C)9 C) N C C) 10 il]~~~~~~~~~~U 1 *f * I.I l ~ tj CJ o Cul C)() C)U)) C) Li0 C) C C-) U Lj~ -) L) 0 C. U U C 0 j 0 C) C) U) U) C) C ) ) C 0 C) ) C N 0 0 C) C) Nl ~,~ ~'l' oo u~ o ~ o C- 4 0 Ou - 0 0 c- ~ 0 0 pj0: U!0 U 01 C) U 0 C)'O U010 a C) C) C) )C C) )C))Um0 C) C) I, 0 r 0' C) C VIC- L U V 0 U Q. 0 0 U 0 C U C C C C C C) 0 C) C) C N - 0'0 0 U L, 1 0f.0 1 C.. ) 0 0 CC) C C), ) U 10 U 00C)CC)) Q C)fl C)Nc 0C U) o l C)'L"C1 C) 0 C) U C-' U) C) 0 0 0 C) C) N 00 N U C N )C). of 0, 0 0' C) C) 0 C C) C U) C C C) U C C N C) N 0 0 C N0 C).1 ~~1 Of'~I ~ ~ 0.0 ~~ C) C) C) C) C-) C) C ) C C ) N N C CD1 C C> O C,' f U O 0 C> C) ) C u O C WI.U -, 0 0 N C) C) ) C) C)) C C:CC)'4'C 0 C) O N N C)'C) 0 0 0 Dco Ico CI C C>~~C~~' ~ o o U ) C )' C ) C C 0 C C) oN N N U C (14 C) L O 001U 0 0 0 0 00QC- t 1 C) N C) C ) C) C)'C-) C) C) )7I C) C) ~ I I ~ I I, 0C C1'' 0 C C ) 0 0 0 0'O 0C C) C C) 0 0 U L() C)' 0P- 0'C -)c CC cc C N' C ) ), 1 0 C 0 C) C) C C) C, 0.0 CONU N0 C ) C) C),''0 C) Of C) N, 0 C) 0 C) 0 0 0)0 C) C) 0 ) 0 4 0 U 0 C N N N co 9~~~~~~~~~~~~~~ C ~ ~ ~ ~ ~ ~ ~ C C) C) C C) -C) 0'O C) C) C) 0? C) C) CD C) cf) C) () r C) ) 4) 4.~~~~~ 4~~~~~~~~~~~~~~ g I C1 gI I I C. I7 gT I I I ~ u v C C C) V 1 C-) C) ) 0 -L 0 C 0 0 C C)0 O C)0 N L) C).i 0 C)\'C'T'T'C0 01' C I C C' C, C) C) C) 0 0, C) C) C) C, 0 C) C) C, C N C).0 0 C) N7 N Na, C'C)0C) 0' C, C) 0 C) a)C C0CC) C 0 0 C 0NC)) NCN C C) C) CCC)CC) C C) C C X~~~~~~~~~~~~~~~~~ o C) C'C 0 0 C C. C) C) 0 C?0 C C) C) C)N N C) 0?' C)-'C -.r- C 0. ~ ~ ~ ~ ~ ~ J l0 I0 IU ~c l J l~ I) 01 ~ I ~I f C)C) C C) -C C0 C. C. C) C' OC) C) 0 C) N O ( C), - 0 C) C) C) C) C) C) C, C), C, C C C) C) C C, C 7 C) C), c0,) C);C 0 C)'C C, 0 C) a C C) c) C) ~~~C C) C)0 C- ) C C) C) 0.0 0 C) C, C) ccCU, ) N C, N - C) C, C) 01 C) C C) Io o oo l 01 00' C 00 - 0 N C)'a C) C a'C CC) 0t 0.C)01 C CO g 00 C) 0( C C o I.1 CI 0 m I o C, \O~~~~~~~~~~~~~~~~~~~~~~~~~~O ui ol ol ol i I o~~l ol o or o! o![ n l~ Yeil - o io io 0.0 C,'CC, cl 04Q 01 CC C) cC ZC N 0. C) C) O C) 0 0 C C ~~~~~~~O C I I,, c, cl N l C" C) N~ Cl )'l ~C C) cl 0I ol c)l ol o. C, 0 1 C01 0 01 T., 2 r Ni 0,0 OC 0.0 0 0 0 C C, - ~ C1 u.; ul C C Lr O-~,, rjC ~ -9~~~~~~ C!! oi I 0: C01 ulOiv - o 0 0 l 1 l I1 i o~~~~~~~~~ CC' C' 0 C: C, C) 0 C) 0 0, C C') C)'1 C) 0 C' 01 0C 0.Cl 0 C - ) C C-.1 C" -)IC) Q C-, C) 0 C) 01 C 0 Q 0 c CC) C. 0: 0 C C!; L: C'O! CC ) C 0) C. CD of *I C)1.C) ~ ~~~~~~~~~~~~ ~~~~~ ~ ~ ~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~, c.) 0 [ o fr, I - I f I of I o c~~~~~~~~~~~~~~~~~~~~~~~~~~~o C, 0 )',',,'..... C,','''i C.. C) ) Of C- ci C).U- C C Ln 0-) U- CCC 0 00 ICD C)1 C 9 C) C.' U (iCC C. C) o~~~~~~~~~ 0~C C-: C)1 t~ O C) C01C. 0 C) C), C. C) Q) C CD; CC C, OI C 0 C 0 0 0, C4C'C4 I ojo] f/ C C C, - C4 C N. )? 00C C C', C CC)'CC-) c) ao~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~o C C C) C! C:,! C, I C C),. 0, 0 0! CD.010 CC C)C O C-~ c C-) C.): C, d C C, C) IC )C)rf,C C- C OO.'''CC ~ N:C CC C 1 -Zg'C., C N C. C ON C, C) C 0101 & 01 ~ C) C) NO 0 0 0 C)' C C. C 0 ci 01 ulCiC CIC 0 01 C C C~~~~~~~~~ r-~~~~ C re IT C,;i Ct~~ ~ ~~~~~~~~~~~ ~ ~ ~ ~~~~~~~~~~~~~~ C C DL: ~ C) C C C) o 0 CC, C ) - C, Crl-'c C C - ) c o 0 C C)~~~~~~~~~~~~~~~~~~~~~~~~: oc C 0 C C CC 0,0, GIC~~~~~~~~~~~~~~( S~~~~~~~I C 9 CI r~ C:! C Oi a C- o 0:~~~~~~~~~~~~~~~~~~~~~,),':C", r.C, C. a IC'C! 0 r- 01 C.) L C, C CD C 01 C) L, C) C, C) 0,CC C C5 C) cc, O r C 0!\ C 01C Cl C C i P. -C, C) -P~0 a C C or at o.. c~~~~~~~ oi) C) C).r L": a, cx c CT C; )!L, U Ci 0: C~~~~~~~~~~~~~~~j COI P~~~~~~C) r - r- C) r, Cr C a' C, - C) C. 0 ) 0! 0 c 0 U' r I C M. C) ~: Cci C'O e

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;. vU o;'ji 0 0 U 0 0 0 0 0 0 0' iool0~ o o 0L I 00 0 U 01 00 0' 0 0 0 V' 0 0 0 0 Li0 0, 0 0I' 0 U I. Q ~. -P ~; 4i U ~ e, 01 0 0 0 00 0 0 0 01U'. m Q j U i 0.;QrU jQt ) C U0V0C0 U I 110 0 0 m U'0 -l 0 e' ~d U U; U+ ~J u~ U C., Q ~JU V. 0' 0l 0 0,J U 0 0.0 0 0 0 0 0 U ) 0V a 0) - U.) C D U a 0C),0 0 0 0 010 0- 0 0 0 U 0) -J 0 0 CC 0 0 0 0 0 0 0 0 0 1 0,: C) U Q U U 0 0 U 0 0: 0 o' o' o 0 0' o: 0 0 o1 o O C 0 00'i'i Q ) C,W U IL ~V -)0 U ci L ) U ) M C D 0C 0 0 0 0, 0. 0 0l 0~ 0 0: U, 0 0' C) 0 L( 0 0 C) 01 0 0- 0- 0 C) 0 M'V 0 4 0) 0, 00 0 0 0 C 0 U 0101oo~~oo 0o0 00 JO: 0: 0 0) o; 0 & ffV 0 C I U, 0C) 0 0 01 o0 0, U U VU o C U C)CUC 0 U VILi %. t_4 11 04 I- - U L l Li! V~i Li]' c oV UC 0 0 V. 0 Vj L. 00.,; V ~ L)' 0 0 Ul C o C) c: 0 C. 0 C C U ) -, C.~~ C-A L) U U 0 C) U U U L.1 U~~C1 0'oV.' C C 00 C CF- C, U Ci C C 0L 0x U'- U l 0 i 0 V 0 V.1Ur V. V. V. \ M V V 0 V U U ~~~LV 0L) C- UC) C- CV)C - 1:- ~~~~~~~~~~~~~~~~~~~~~~~~v C1 ~ 0 t.` C C1 C - C) 0C-) Cl cL- C C C1 V. Q C C 1. CJCC CC V. C C C'V oCl CIV.cV., 0 V. C,; C- CD C) 0 C-, C" C) L: C C C) C., c-) U C,CI CCU C V. C Cl o. -' CC C. ~.~ Q; C.': C' C C~ C ( D C, C-CC C C. C.. C. C C., ~ ~ C,-C; C; O' C 0 C:-', C.' C'. C, VC C I2 C C Ii C ci C1.~ ~ C, CCCCC *~~~~~~~~~~~~~'' I~~~~~~~~~~~~~~~~~~~~~~~~~~~~C C' C)C C'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~. cV:. 1, Mi C) C..I C C C C 20 00 0 C ~ C -'C C C: 0 0 0 0 0 0 C V. V. V. C C, V:. C r-4~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~ <!. C:':211. C; C i C) L o'' 0! C C: C, C, C-I C- C I C, C I L~~~ ~ ~ ~ ~~- C C, C'i C CC C~~' C~~ % C~~ C, C C C' C: C) C — i C C.C ~.... (-w;~ ~ ~ ~ ~ ~~~~~~~~~~~~~~~~~~~~~~~~c. C-C, C Lc C; U' -,.! ~,,U C. [:: C, C, C" C[ i (-:~ C% [ CC. L ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~C.: r 71 C L) Q ~~~~~~~~~~~~~~~~~~~~~~L~~~~~~~~l. c~~~~~~~~~~~~~~:?i C,~." ~, - C Ol C C~~~~~~~, C C~~! CrC!C C C — c.C-i C, 8, Cc-C6.CI -, C: C C — i C' C, C C. C —' C' C l C. C.C, _;C C'C' Ci C C C C C C 0~ C' C 7.1 C C:,. C' C-; C- ~ C.~ ~ ~ ~ ~ ~ ~~~. j I I ~ ~~ C.' 7 C.- C' C C;, C ci C,: C". L- C L. C C -L-, C, C Q-~ ~L.C ~ C,. r.C C! C, Cr Cl~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ir ~~~~~~~~~~~~~~~~~~~C L~m ~ C C.C CC C:!r. 8', c CiC.C C CC, C C C C C, C 17 C1 CD, C,.)~C C: C_ ~. 2: C. C C',~~~~~~ ~~ ~ ~~~~~~~~~~~~~~~~~~~~~ C,.C,.. C! C c! C (.-. el'~ I....~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~, O'0 & C:0~C: L C O' C, C), C ~, C; C, Ci U. C! ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~~,.. if...,~ C,. o c o q c,

171 uL~.000000, FREQ =.000000O, O _ 0 = I ~.0~5.0000.C77.2770.04.2803.~C85.0000.00~83.0985.0000.0978.0081.0000.0080.0084.0000.0084 DEOLL.OCOOO FREQ = ~10.782683, J = 2.0S2C.0C.~29.2I4~2.0005.2892.0074.0000.0077.0920.0000.0929.0075.0000.0076.0079.0000.0080 0600 ~.000000, FREQ 40.78~3369, J = 3.0460.-C435.02.C.7349.0007.161.0120 OOC.0467.0000.0002.0039.0000.0023.0039.0000.0015 o)LLLO =.OREoOO, 6660 90.647484, J 4 _______ ___.00.000-.14~16.001I.00.2624.-CC0.0000.0436.0000.2397.1479.0166.0538.0122.0158.0538.0121 Uo~~LLC~,=000000,,C~ ~4640 =.....A. J = 5 ~.CCOO.0Q1.0~7,33..0000.1644.00O.0000.C005.0000.3140.1393.0435.0983.0087.0449.0979.0095 0000C.0,fkEQl~ 346.50~5070, J = _6.__.1010.301 04l.~51'.0CC.0261.0130.0000.1847..001.0000.0149.0000.0210.0135.0001.0206 =E L.000000, FRol 635.249420, J= 7__.1476.A19oo 00ic0.00CC.00 11.0 OO 0CC.0095.0000.0000.0041.0006.0450.0045.0007.0459 iALL u )C:JCfKQ 829.75576 -J= 8.Ccou CO.O.0011.0000).00,00.1575.0000.2148.0241.2572.0442.0000.2559.0446.0000 OLLL v0~ C66o0k 1067.801300, 0 9 _ _ _ _ _ _ _ _ _.33-4o04 0%ce.-3.0001.0131.0073.0054.3606.0029.0202.0031.0075.1308.0120.0073.0079 o0LL00 0 0,00I FRCO 1088.146973, 0 _ 10 ______ _____.C060'3.0010D.2 t'ii.000,7 llC3 C)0043.0014I.0144.0923.0065.0521.0062.0021.1662.0000.0018.3102 L)0 L C030 = 1171.993027, -J1.05-~2.2 71 C. C C00 C.0000 5020.2767.000~4.0000.0001 -.0001.0004.0QO1.1593.O001.O001.1338 uLLO = 6660J O =HE 1189.4934481, 0 12 __- -__.00C0C8 C00 C.0C1O O.0005.0002.0004.55C2.0005.1010.1364.0373.0003.0641.0353.0006.0705 060 L= C. ~ 000 FR66 1321.556519, J= 13.0oLI2- 10 J,2.0011.00C1.000 1o3.0613.0000.0186.0000.0000.0039.0001.2635.0037.0001.2580 E L00, 62 R =, 1426.i702667~ ___ _ Js 14.00013.0000.~-i CC.0000C.0003.00.0000.2108.0003.0090.289C.0031 +.0000.1108.0046.0000.1129.00u0 6604 1454. 781 876,-.J= 15.0~227 C0001.000 1.0000.0000.0009-,000P~40317.0000.0000.2873.1842.0004.2867..1842.0006 ALLO FREQ00, 66 = 854603 - 16.0020.002.0 000 CC).00 C.00.00CC.0010.0012.0000.0639.0000.0000.1619.3050.0000.1619.3050.0000 0000( =.00C')00, IRQ 2.9.27.I2149-28~jL ___17.0000 C000.001~0.00C0.0000L.000C.0000.0000.0003A -QQO-0.1119.0000-.1406 —.3-031.0000.1408.3031.0000 000 =.00-00 FREQ 2984.3245241 __ _ 18.031.3.C590.0000.0000~.00CC.0000 1.301.4.65.OC.07.00.00.01.0006.OQQ- 001 0006.0001

172, DELC =.349066, REQ = 8.051599, J = 1.1832.C000.0008.5617.0003.0241.0150.0000.0004.1826.0000.0006.0153.0000.0003.0153.0000.0003............. OtLC =-. 349066._ FREQ = 40.190042. J =2.C465.C486.0004.0808.7294.0067.0161.0120.0002.0472.0001.0001.0042.0000.0017.0038.0000.0020 DELC =,.349066,, _ _ __REQ: 57.235491, J 3.OC41.CC L.__.0296.0130.006 ~_.3.0001.0000 008.0599.0299.0036.0130.003130.004328 OELC =.349066,,FREQ,.=- 126.970289 J = 4.0021.C066.-674.0023.0001.0002.0008-.0002,.0597_.0004.2150.2714.0146.0482.0244.0143.0482.0242 - E.L.. =,,....490.6...__.__......FREQ = 325.212990. J 5.OC87.C066.11i.0029!.0008.1537.0019.0001Qo.0141.2639.1707.0374.0825.0114.0388.0822.0122 ~~~~~~~.0C87.0019. 00.0141 6392.1070.037-4.. OELE =.349066, FREQ 347.996250, J = 6.CS33.2883.3002.2422.0488.0004.0238.0126.0015.1737.0148.0023.0210.0090.0199.0196.0091.0195 DELC =.349066, FREQ =,,633.642563, _ J = 7.1446.4-15.COOI.0949.1994.0000.0010.0191.0004.0091.0036.0011.0074.0011.0435.0078.0012.0442.ELC. =., 4906., [__ __F = 833.392937. J 8.OC55.0067.0004.COOO.0003.0010.0006.0007.*1569.0014.2104.0251.2510.0429.0020.2495.0434.0023 OELC =.349066, FREQ.= 10,56.581940____ J = 9.2625 0CC9..0942.0005.0035_.~000_.1,0,06.0075.0029.3507.0013.0120.0052.0065.1272.0118.0062.0673 OELC =.34906b, FREQ =. 1095.884628, ____ J=.10.1319.0024.23o1.COO1.0007.0004.0061.0024.0253.0938.0137.0671.0070.0026.1627.0030.0024.2423 OELC =.349066, FREQ = 1170.366119 J = 11.C283.C084.COCC.0004.0093.0000.3251.2533.0426.0009.0072.0071.0035.0001.1546.0031.0001.1360 uELC =.349066, FREQ = 1193.757462, J = 12.0145.~CCC1.0006.OO0.0007..00_04.0257.0215.5008.0079.0891.1218.0328.0007.0752.0312.0010.0760 OELC =.349066, FRE -= 1320.108170, J = 13.OC13.1165.0002.0011.0001.0000.2704.0626.0003.0186.0006.0006.0041.0001.2615.0040.0001.2561, __LC= -,,_.349066, FREQ = 1426.406143. J = 14.OC04.00C03.2578.C0000.0OOG.0003.0000.0000.2081.0003.0085.2900.0032.0002.1119.0047.0002.1140 DELC=.349066,. FREQ = 1454.593292, J = 15.0219.0011.0003.0001.0001.000C.0000.0008.0006.0322.0000'.0003.2868.1843.0004.2862.1842.0005 DELC =.349066, FREQ = 2855.454742, J = 16.OCOO.00C0.0000.0000 O.000C.0000.0010.0011.0000.0638.0000.'0000.1619.3050.0000.1619.3050.0000 OLLO.349066, [REQ = 2927.107849. J 17.0000.0000_.0000_.0000.0000.0000.0001.0002.0003.0000,.1118.0000.1408.3030.0000.1408.3030.0000 DELL __.349086, - F REQ = 2984.306854, j 18.0312 C590.0000.0000.0000 00CO.3013.6057.000C.0007.0001.0000.0002.0007.0001.0002.0007.0001

173 DELC =.698132f FREQ - 23.529515, J = 1.1192.0040.0020.5896.0598.0764.0066.0010.0008.1176.0002.0014.0098.0000.0010.0098.0000.0010 V)ELC =.698132t FREQ = 40.676315, J = 2.0899.0451.0017.C125.6571.0491.0218.0109.0008.0899.QOOO.0010.0080.0002.0021.0077.0002.0021 DELC =. 698132, FREQ= 62.253693 J =3 3..0111.0135.0010.0653.0205.6761.0005.0008.004C.0091.1272.0024.0073.0259.0010.0075.0259.0010 DELC =.698132 FREQ = 196.157688 J = 4.0231.0237.1682.0016.0000.0961.0058.0005.0467.0104.2564.1730.0183.0581.0209.0184 ---------'~~~ —--..1730.0183.0581.0209 -0184.0580.0208 DELC =.698132t FREQ = 351.826778t J 5.0937.2594.0271.2265.0451.0053.0226.0115.008S.1857.0082.0056.0231.0098.0149.0257.0096.0173 DELC -__ _.698132, FRE Q = 356.501324, J = 6.OC17.0045.2062.0045.0002.0949.0002.0004.0066.0052.1609.2810.0345.0607.0240.0319.0606.0219 DELC =.698132, FREQ = 629.540115,. J__ = - 7.1369.4030.0004.0978.2016.0002.0008.0179.0009.0076.0130.0045.0151.0022.0398.0154.0023.0405 DELC =.698132, FREQ = 843.337112, J = 8.0217.0222.0014.0000.0007.0008.0027.0026.1531.0071.1982.0290.2342.0396.0067.2321.0401.0078 __ DELC._698132, FRE _ 1037.065231, J = 9.2551.C161.0952.0006.0044.0000.0089.0090.0020.3703.0043.0038.0100.0059.1125.0163.0054.0803..D.ELC __=.698132, FREQ = 1106.164978... _ _ ___ =J 10.1441_.0037-.2392.0000.0006.0006.0055.0036.0466.0552.0255.0907.0101.0021.1580.0062.0020.2063 DELC =.698132, FREQ = 1167.317200, J = 11.Q164..0306.0009.0004.0085.0000.2957.2274.0778.0045.0115.0089.0058.0002.1580.0055j.0002.1476 DELC =.698132, _ FREQ =_ _1203.849716 _ J = 12.0342.COC1.0002.0000.0011.0003.0497.0402.4474.0206.0732.1024.0269.0014.0896.0259.0016.0851 DELC =.698132, _. FREDQ = 1315.903290 t. = 13.0013.1130.0006.0010.0001.0000.2767.0664.0014.0187.0023.0024-.0047 -.0001.2560.0046.0001.2506 DELC =.698132, FREQ = 1425.322021, J = 14.0006.0011.2'547.0000.0001.0003.0001.0000.2005.0002.0070.2929.0035.0006.1153.0050.0006.1174 DELC =.698132, FREQ = 1454.048950, J = 15.0198.COC9.0012.000 0 1 0 000.00000 0000.0008.0023.0336 0001.0010.2857.1845.0002.2849.184** 0004 DELC =.6981 32, FREQ = - 2855.369812, J = 16.0000.0000.0000.0000.000C.0000.0009.0010.0000.0637.0000.0000.1620.3051.0000.1620.3051.0000 DELC =.698132 FREQ = 2927.062317. J 17.0000.0000.0000.0000.oO.0.0000.0005.0007.0003.0000.1116.0000.1406.3028.0000.1407.3028.0000 DELC =.698132, FREQ - 2984.261627,.J - 18.0311.0590.0000.0000.0000.0000.3011..6053.0000.0006.0002.0000.0002.0009.0001'.0002.0009 i0001

174 DELC = 1.047198, FREQ = 30.018087, J = 1.0066.0373.0002.4144.4911.0289.0007.0084.0000.0063.0020.0002.0008.0000.0012.0008.0000 *0012 DELC = 1.047198. FREQ = 53.026761. J = 2.0654.COCO.0037.0008.0853.6546.0129.0009.0000.0668.0598.0011.0097.0137.0008.0098.0137.0008 _ DELC = 1.047198 ___, F RE _= 62.692419. J = 3.1215.C472.0029.2713.1633.080C.0109.0043.0097.1138.1045.0039.0124.0174.0037.0122.0174.0036 DELC = 1. 047198_ FREQ;_ 228.475094, J = 4.0682.0384.0503.0003.0004.1956.0148.0006.0226.0399.2951.0672.0227.0661.0144.0229.0660.0145 OELC = 1.047198. FREQ = 358.495075, J = 5.0725.2270.0049.2063.0355.0000.0167..0110.0090.1490.0553.0058.0476.0419.0139.0476.0418.0142 DELC_ 1.047198, FREQ = 424.086063, __.__ J = 6,,OC59.C002.331 7.C031.0043.0388.0008.0001'.0379.0415.0346.3621.0160.0185.0352.0160.0183.0350 E _ LC - 1.04719,,,, FREQ =, 624.666939,_ __ _ _ J = 7.1289.3786.0019.1016.2037.0004.0006.0163.0007.0049.0254.0117.0227.0031.0365.0229.0031.0370 DELC = 1.047198. FREQ = 857.223938. J = 8.0497.C369.0048.0000.0009.0006.0066.0047.1428 _0216.1795.0393.2102.0350.0110.2072'.0357 0t134._.....3.......'......._0. _,0L - 1.047198,, FREQ = __1017._309120, J = 9.2298.C281.0977.0006'.0055.0000.0071.0118.0052.3800.0138.0014.0183.0056.0937.0255.0050.0709.....___ _..DELC= 1.047198, FREQ = 1114,.016968, J =10.1451_...0.070__.2389.0000.0006.0006...0029.0069...0731.0214.0Q372_.1103.014.0013.1471.00.0013.1814 DELC = 1.047198, FREQ = 1164.082306, = 11 08 2_.0_2_ 0_68.003.007.0000.22.2010003.0.005.0071..050.009 000.1753..0059.0004.1693 DELC = 1. 04718_ FREQ=_ 1216.838654, J = 12.0483.C016.0043.CO001.0011.0001.0666.0499.4120.0278.0606.0875.0226.0020.0984.0221.0021.0931 DiL-C_ - _C 1.047198,_ _ FREQ = _1309-363861 __ J=' 13.OC1.2__.10.36__0008.0010..00_.OO.0000.2855.0723.0048L.0190.00 6_.0055.0054.0001.2479.0054.0001.2426 DELC = 1.047198. FREQ = 1422,892883, J = 14.0C09.0022.2489.0000_.0002. 0003.00.000.1903 0002.0051.2976.0036.0009.1208'.0051.0010.1227 DELC = 1.047198, FREQ= 1453.198608, J= 15 1.0166.0007.0020.0001.0001.0000.0000.0007.0042.0357.0002.0016.2843.1851.0001.2834.1850.0003 _DELC = 1.047198, FREQ 2855.239471, J = 16.0000.0000.0000.0000.000C.0000.0008.0009.0000.0636.0000.0000.1622.3052.0000.1622.3052.0000 DELC = 1.047198,. FREQ = 2926.9934089 J = 17 -.0001.0000.0000.0000.0000.0000.0010.0015.0002.0000.1114.0000.1405.3024.0000.1405.3024'.0000 DELC -= 1.047198, FREQ = 2984.192322, J 18'.0309.C590.0000.0000.0000.0000,3008.6047.0000.0005.0005.0000.0004.0013.0001.;0004.0013.0001

175... ~C='9 __62_3..__ 27.7138902 ='.__.03 __ _.O_. _.3.11.......36.....0_2_11!........00-22.0106.000.0003_0Q5_ 0 000 _6.0000.0012.0005.0000.0012 DULC = 1.396263, FREQ = 48.971233, = 2 59 -.0033 --,.0-0, 0o8-,000. 17-51l —-.0000'.0068.032Z.0008.0069 02 08.0004.0270.0002.._1163.0002.5941 0020...0000.0068 032.0008 0069 0322 0008 DELC 1..396263,._ FREQ... 87.414615........ J: 3.1474.C369.0098.2799.1462.1415.0141.0037.0089.1397'0283.0053.0114.'0031,0047.0113.0031.0046 DELC - 1.396263, FREQ = 239.553902,.. J= 4.1212.0313.0112.0036.0041.2245.0230.0003.0077.0716.2702.0363 _.0236.0605.0134.0238.0604.0134 ELC: 1.396263. FREQ 367.180813. J 5.0714.2013.0057.1764.0248.0005.0149.0103.0112.1470.0654.0112.0619.0555.0124. 0620.0555.0126 DELC = 1.396263,... FREQ 501.205795, J = 6.OC18.01C8.3329.0079.0142.0161.0000.0006.0666.0678.0046.3347.0183.0105.0423.0183,0104.0622 DEL 0 = lo.396263,. FREQ: 621.23.4978, -. J: 7.1262.3459.0093.1030.1998.0009.0006.0143.000.0009.0383.028.0255.0029.0373.0257.0029.05378 DELC = 1.3962631 FREQ = 872,724686 J = 8.0902.0429.0166 _OOCO.0006.0004.0130.0062.1173.0501.1546.0639.1807.0295.0'120.1761.0.305.0156. DELC = 1.396263, FREQ = 1002.517670, J: 9....1898.0476.1091.0006.0067.000C.0042.0164.0189.3606.0304.0028.0312.0060.0746.0400.0051'0559.............. DELC= 1.396263, FREQ 1118.454651, J = 10...1472.0141.2240.0001.0010.0006.0004.0153.1028.0043.0476.1236.0201.0007.1280.0147.0009.1547 DELC = 1.3962639 FREQ = 1160.974121 J = 11 __.0024.0310..0249.0003.0071.0000.2399.1759.0828.0086,0069.0010.0047.0006.2072.0049.0006.2015.DELC._ 1.396263, FREQ= 1231.122711, J = - 12_..0558.'0063.0147.0002.000S.0001.o0915.0601.3804.0262.0516.0763.0198.0023,0978.0197.0023.0942 OELC -- 1.396263 FREQ 1301.292618,_. __=. 13.OC08,c895.0004.0008.0004.0000.2912.0782.0149.0199.0065.0109.0058.0002.2395,0059.0001.2369 DELC 1.396263. FREQ = 1418.310654. J = 14,0013.0034.239C.0001.0003.0003.0002.0000.1796.0002.0032.3035.0032.0011.1288.0045.00.130 0ELC= 1.396263, FREQ = 1452.127441, J = 15.0128.0005.0023.0001..0001 00OC C00O0.0006'.0055.0383.0002,0018.2833.1861.0001.2825.1860.0001 DEEC: 1.396263. FREQ = 2855.0783399 J = 16.0C;00'OCO.0000.0000.0000.0000.0006.0'007.0000.0633.0000.0000.1624.3052.0000.1624.3052.0000 DELC 1.396263, FRE = 2926,908569, = 17.0001.0000.0000.0000.000C.0000-.0017.0025.0002.0000.1111.0000.1403.3019.000.1403.3019.0000 ELC = 1.396263, FREQ = 2984.1078809 J = 18.0307.0590.0000.000 0 00.0000.3005.6039,000C.0004'.0008,.0000.0005.0017.0001.0005-.0017 o0001

176 DELC = 1.745329s FREQ - 22.3550151 J 1!.0000.0672.0000.2442.6283.0280.0022.0119.0002.0000.0138.00O.0007.0003.0011.0007.0003.0011 DELC = 1.745329. FREQ = 39.812308. J 2.0000.0450.0000.2282.0010.4207.0024.0007.0033.0000.2118.0000.0064.0361.0009.0064.0361 o0009 DELC = 7.745329: FREQ = 110.993520, d 3.0870.0413.0110.2363.032.3562.0068 032.0086.0840.0317 0037 0067 0030 0039.0066.0031.0038 - DELC = 1.7453291 ___ FREQ= 250.5492659 J 4.1725.0CI1.5 _ L0_365.0__7.182.0285.00 013.0 3.901.2328.0295.0261.0567.0158.0263.0565.0158 DELC = 1.745329, FREQ = 379.411369, - 5.0860.1822.0075.1359.0131.0024.0160.0095 _.QL4___.1_6_18 _._6,00.0188.0725.0610.0130.0727.0610.0132 — DELC = 1,745329,..... -_FREk_=_. 557.795677_,........J 6..0C17.0632.2558.0260.0511.0082.0004.0029.0873.1165.0013.2255.0339.0105.0359.0338.0103.0359 DELC= 1.745329t FREQ= 622,260_52.... 7........ J =7.1368.2762.0408.C910.1685.0020.0011.0109.0071.0060.0451.0738.0182.0013.0507.0183.0013.0510 DELC = 1.745329, FREQ = 890.2939159 J - 8.1298.04C6.0457.CO01.0003.0003.0204.0071.0721.0813.1269.1139.1483.0233.0095.1420.0246.0137 DELC= 1.745329, 1..REQ = 997.641861,........ J = 9.1361.0714.1294.000 _.,0073.000CC.0012.0227.0503.3087.0505.0054.0460.0070.0582.0571.0058.0424 DELC= 1.745329_, FREQ = 1118.482681,.._ -. 10.1514_.0__266._1_910._0001.0017.0005.0014.0303.1302.0025.0557.1309.0259.0006.1029.0199.0008.1275 DELC= 1.745329t- FREQ = 1158.353683, J = 11.0C02.0254.0616.0001,.0057,00_0.1944.1351.0696.0055.001...000_ 3.0028.0006.2514.0031.0005.2405........DELC - 1.745329, FREQ = 1245.339050,... J =__ 12.0572.0179.0311.0004.0009.0000.1498.0820.3328.0' 172.0474.0644.0192.0021.07816.0192.0022.0777 DELC = _..745329 - FREQ 1_.93.325867j J 13.OC01.0678.000 0.0003.0006.000.000.27 21_.0761 0482.0219.0062.0223.0048.0003.2379.0050.0003.2355 DELC = 1,.745329. FREQ = 1410.693222. J = 14 o0C19.0046.2227 0001.0004.Q003.C002.0000.1695.0002.0016.3101.0024 o0009.1402.0033.0009.1408 DELC = 1.745329, FREQ = 1450.964279 J = 15.0C89.0003.0019.0000.000C.0000.0000.0004.0059.0409.0001.0014.2828.1875.0001.2823.1874.0000 DELC = 1.745329, FREQ = 2854o905731, J. 16.OOO.0000.0000.0000.0000.0000.0004..0005.0001.0631.0000.0000.1626.3053.0000.1626.3053.0000 DELC = 1.745329. FRED = 2926.818115, - J 17 J.S.__L.OO_. _ ~._ O.oo 0000.000oo000o.0025.0036.0001..000o.1108.0000.400oo.3013.0000.,1401.3014.,0000 DELC = 1.745329, FREQ = 2984.0177921 J = 18.0305.0589.0000.0000.ooo.0000'.3001.603.1.0000.0003.0012,oooo.0007.0022.0001 oo.0006.0oo22.001

177 OELC = 2.094395__ FRE =6.5 50159g J = 1.0000.09C8.0000.1672.6488.0376.0016.0134.0004.0000.0342.0000.0007.0015.0009.0007.0015.0009 DELC = 2.094395. FREQ = 29.672337. J= 2.0000.0536.0000.3565.004Q6.2545.0036.0006.0023,00.00.2341.0000.0061.0383.0006.0062.0383.0006 DELC = 2.094395t- FR 1 123.844499. J = 3'.0360.0380.0093.1579.0635.5940.0020.0024.0063.0397.0331.0017.0030.0031.0020.0030.0031.0020. Lo~.i;_= 2....].__Q_395, _]......________F a~RIQ = 267.47 5544' 4 9_28__,0001_....0_000.....1_05_......0_3_9?....._0939....._2_______0_0__01....00~0___._5 _._ 0__ ___,0_.0,0605.0186 9.0318.0603.01_8 DELC 2,094395' FRED = 39'8.06720, 1= 5.1'i.'6410102~_40dQ030.Q057.02.03...OA.016.1859..0.24.0317.0775.057.0180.076 57.0181 DELC = 2._Q9_3:_95 _ FREQ = 580.38 575, J = 6.0051''.1549.1445.0574.1128.0047.0010.0066.0806.1571.0164.0977.0528.0111.0167.0527.0109.0170 _L_C__ — 0 2........2d _ -_ F._~ tQ 2...........9...6_R = __ —_......b2 6,./t 7Q _9(_L. _ J = ~.__.(:._3......[_7_C3.0786,0649.. 109.. 0036.0034.0063,0338..0382.1073.009.....0005.0687 DELC =.2.094395, FRED = 914.296265. = 8.L40.J_4_O_3____,O, 0819.0001,0003.0003.0266.-009.0273.0813..079,81Q _.1232.0180.0_18.1148.0195.0150 DELC_.'=..._,_2.094395,.. fEQ = 1003.691696,. —--- 9 E0885,C865.1627.0002.007C -.0000.0001.0285.0939.2523.0620.0076.0527.0076.0447.0670.0060.0326'DELI 2.094395t. FREQ = 1113.650833,.J =. 10...1585.C430.1483.0001.0028.0003.0049.0463.1417.0092.0611.1361.0316.0008.0807.0247.0012.1087.DELC: 2.094395. FREQ = 1156,2280589 = 11.0C18.C170.1109.0001.0040.0002.1354.0880.0589.0015.0009.0030_'.0014.0005.2985.0016_.0004.2761 DELC = 2 _ 094395....FREQ 1= 256.101410,....J = __ 12.0497.0443'.b413.0008'.0014.00CO.3068.1389.2113.0045.0494.0381.0215.0016,.0340.0216.0015.0335 DELC = 2.094395 _..FREQ = ]289.8688661.. J -. 13......0025.0312.0147.0002.0007.0000.1627.0433.1628.0247.0015.0514..0014.0007.2476.0016.0007.2523 DELC = 2.094395, FREQ = 1399386490. 14.031.0055.196200.OOL.0005:.0002.0002.0000.1588.0001.0005.3172.0015.0006.1570.0019.0006.1558 DELC= 2.094395, FREQ 1449.867569 J = 15.0053.0002.0013.0000.0000,0000.0000.0003.0058'.0433.0001'.0008.2826.1889.0001.2825.1888.0000 DELC = 2.0943954,... FREQ = 2854.741730, J.= 16.0000' O00O.0000.0000.0000.0000.0003.0003.0001.0629.0000.0000.1628 -.3054.0000.1628.3054.0000 DELC 2.094395, FREQ 2926,733582,' = 17.0002.0001.0000.0000.000C.0000.0031.0046.0001.0000.1105.0000.1398.3008.0000.1399.3009.0000 DELC = 2.094395, FREQ = 2983.934021, J = 18'.0303.0589.0000.0000.0000.0000..2997.6024.0000.0002 -0015. -— 0-001.0008.0026.0001.0008 -.0026-.0001

178 DEL.C 2.443461 _ FREQ = 10.218006, 1l.0000.1231.0000.0922.64.57.....0365.0009.0147.0Q004.0000.0747.0000.0006'.0048.0006.0006.0048.0006 DELC = 2.443461: FREQ = 19.830628. d - 2l.000__.0513.0_000..3..0279.118.0049_.0005.0011.0000.2328.0000.0060'.0378.0003.0061.0378'0003 DELC - 2.443461, FREQ=.4258 J = 3.0109.0231.0070.0769.0300.7976.0004.0013.0037.0182.0222.0005.0013.0020.0007.0013.0020..0007 DELC: 2.4434b61., FREQ 284.47_25910,__ ___ J= 4.1734.0070.0002.1867..0557.0371.-0234.0022.0000Q _._ 371.1968.0215.0397.0709.0188 -.0403.0706.0187 - DELC = 2.443461, FREQ.= 425.959740. 5.1742.1456.0129.0352.0003.0073.0268.0072L.0179._1890.._0218.0525.0762.0490.0295.0760 e0490.0295 DELC 2.443461, FREQ = 581.223373t.... 5 = 6'.0007.1537'.1092.,0537.1086.0055.0022.0064.0853.2442.0204.0532.0576.0103.0102.0581'0102.0106...DELC =...2.44_3461, FREQ 620.814392,......... 7.2010.1580.0507.0724.1158.0032.0052..0057.0343,0983.0428.0653.Q124.0004.0612.0115.0005.0612 DELC = 2.443461, FREQ = 943,983627, J = 8.,1218. 05_6. __.821..0001.0008...0002.0326.0156.0062.0492.1061.2260.1154.0153.0252.1044.0172.0253 DELC = 2.4434 6 F REQ _.. 1016.966103 i. J = 9.0577.0790.2244.0001.0056.0001.0011.0290.1425.2205.0550.0105.0432.0069.0323.0627.0047..0246 D___ OELC: 2.443461.... REQ = 1104.667999,__ ~ J = 10.1754.0618.1058.0001.004C.0002__.0060.0571.1239.0164.0659.1480.0377.0012.0644.0290.0018.1013 DELC = 2.44346.L, FREQ = 1154.083282, = 11.0034. 0089.1592.0000.20.00._ _02.0_,27 0446.0581......._000021 _02_0003.0043.0009.0003.3413.0008.0002.3026 _ _ DELC 2.443461s.......FREQ_=_..1 257,_5_72433____ __ - 12.0323.0631.0225.0010.0024.0000.4876.2018.0597.0000.0488.0095.0218.0068.0137 90221.0008.0119 DELC = 2.44344j 6FRE_ = 1298.073700,__ = -13.0105.0035.0711.0000.0004.0000.0329.0065'.3204.0190.0003.0798.0001.0011.2183.0001.0012.2348 0ELC = 2.443461t FREQ = 1384.423.798. =, 14.0058.0063.1541.0002.0007.0002.0001.0000.1408.0000.0000.3282.0009.0004.1833'.0009.0004.1778 D-ELC = 2.443461, FREQ = 1448.985443. J. = 15.0024.00C1.0008.0000.0000.0000,0000.0001.0055.0452.0000.0003.2825.1900.0002.2826.1900.0000 DELC = 2.443461. FREQ = 2854.607513..J 16.0000.0000.0000.0000.0000.0000.0001.0001 O0001.0627.0000.0000.1630,.3055.0000.1630,.3054.0000 IE.C = 2.443461- ~ FREO = 2926.664429.. Ax = 17.0002.0001.0000.0000.oo0000.000.0037.0054.0000.0000.1103..1397.3004.0000.1397.300,000o DELC 2.443461.'= 2983.86, 295 J " 1'8 5oo.0029.000.0oZ.os89.0oo. oooo.oo oo, ooo.z0.1o8.oooo.,~Qoi.oo,00oo;o, -.0030_o0o_.__J ~:O,:-o~`.oo

179 DELC = 2. 79 27 _ FREQ = _ 4.10_7531 =.0000 1539.0000o.0584.6010.0163.0006.0151..0002.0000.1318.0000.0006.0106.0002.0006.0106.0002 DELC = 2.792527.. FREQ = 11.156023t J = 2.0000'.0429'.OQOp.5528.0876.0246.0057.0010.0002.0000.2050.0000.0059.0341.0001.0059.0341.0001 -__O_ ELC_= 2.792527, R __- -. R 127.048115, 3.0021.0069.0056.0203.0079.9343.0000.0004.0022.0104.0070.0001.0007.0006.0001.0007 06.0001 DELC = 2.792527 - FREQ =.295_,765736. J = 4.1469.C197.0001.2310.0613.0079 __.0188.0035.0000.0088 942.0171.0476.0802.0172.0484.0800.0172 DELC = 2.792527. FREO = 459.661556 = 5,202 _..14_ __.Q1033.0083..0_075.0044 _.03420_.067.013_.11 083.0845.0741.0435.0486.0736.0436.0485 _._... DELC - 2.792527, - _FREQ =. 568_.,306_,J = 6 -.0067.C593.1199.0157.0374.0107.0033.0025.1146.4588.0056.0317.0490.0067.0104.0506.0066.0106 DELC_= 2.792527.2. FREQ-= 610.391602, __ _ J = 7.2CQ9.2323 0C99-. 1121.1815..0010.0036.Q086.0084.0331.0641.0210.0237.0008.0382.0218.0008.0384 DELC = 2.792527. FREQ = 968.697830 J =8.0964_~ _08_580_____37......0000.002. 0002 1..0404..0276 0_Q26OS17.1186.2323.1234. 0151.0405.1092.0174'.0359......._DQELC.79252.? --..:A.._EQ 3 3569 = 9.0302.0430.3235.0000.0029.0002.0017.0183.2060.2167.0299.0106.0190.0049.0251.0469.0020.0190.~~~~~~~~~~~06. 0 2 ~ I _ DE L.2 —. 79 2527, 9FBE _0___.__6 _ F._R_9. = ~ 10,2lO_._0862.054.. _.0056 _001.4..._.80__6Q_65______0_8__ _0164.0752.1736.0473.0018.0503.0340.0027.1010 DELC = 2.792527. FREO = 1151.926331, { = 11.1Q2.O0Q 2 1.0002.0025.0008.0091.3748,0003.0001.3199 OELC = 2.792527, FREQ = 1254.2857829 J = 12.0230.0604.0056.0010.0033.0000.5590.2295,.0097.0001.0472.0024.0207.0006.0082.0210.0005.0079.. DELC = 2.792527, FREO = 1313.819702, J = 13.0078.0000.1523.000d.0002.0001.0035.0004.4119.0111.0007.0631.0004.0012.1572.0003.0012.1885 DELt = 2.792527, FREQ = 1367.454880. J 14.0114.0067.0846.00020.0008.0001.0000.0000.0921.0001.0003.3609.0007.0002.2289.0003.0002.2127 DELC = 2.792527. FREQ = 1448.420090. J = 15..~~~~~~~~;.b0006.0000.0006.0000.0000.0000.0000.0000.0053.0464.0000.0001.2824.1907.0002.2827 1907.0001 DELC = 2.792527. FREQ = 2854.519226,. - 16.0000.0000.0001..0000.0000.0000.0000.0000.0001 a0626.0000.0000.1631.3055.0000.1631.3055.0000 - DELC - 2.792527. FREQ = 2926.619537.. = 17.00co02.o01.0000.0000.00.0000.0041.0059.00 0. 1101.0000.1395 *3002.0000 *1396.300o3,000 OELC = 2.792527.. FREQ 2- 83.821381...0300 ~05:89'.0000 7.. 0 ~, 600000 000.0000'9 6014.ooog jQQ9 019 0-0'..'-9':U.'1.. ~'OO.3-old.'....'": [iXl

18o DE CC 3. 14 1593, FREQ =.000000. J.0000.1909 J.0000.ooOO.5102.0000.0000.0160.0000C.0000.1908.0000.0000..0160.0000.0000.0160.,0000 ULL 3.141593, FREQ.-=.5"60.41# _____-_.0000.0144.000C.6257.1241.0000.0065.0005.0000.0000.1570.0000.0065.0295.0000.0065.0295.0000 0110 3.141593, FREQ = 126.501095, J 3..OCOO..0000.0051.0000.0000.963t.0000.0000.0017__.0064.0000.0000.0006.O00.0 0000 06.00.00 UELC,= 3.141593, FKQ= 299.328583, 2 4.1366.C251 C0020.2438.0621.0000.0172.0039.0000.0000.1935.0155.0508.0836.0165.0516.0833.0164.. 14 15 9-, U RE = 479.1663044, j =.2744.15b7.0-JC.0016.018S.00CC.0410.0074 O00CC.0000.01038.1171.0765.0455.0670.0755.0456.0669 JE. 14 i5 )3, F RE = 554.690308, 2 = 6 o.0CC01.0..o. 001 ol.0166.0000.00010.1347.6352.0000.0000.0395.0006.0006.0417.0007.0006 OE(.141o913, [KLQ = 606.9067-76, 2 = 7.1843.265.176.92.000CC.006.09)9.0001.0001.0724.0103.0303.0015.0281.0282.0015.0285 UELO C 3. 1,t169o,) - 61 = 979.0,-68832, 2.0648.102 9.0 C3.0000'.0029.0000.0C4 47.OoO.0327.0029. 12 70.2 266.1299.0154.0459.1141.0O179.0390 DECO F K14L,, 91 1040.743942, 2 9.0021 c01c9.3oe.000c.00ut.00)03.0'017.(0(56.2101.2255.0095.C114.0007.0036.0259.0358.0002.0160 0110C 3. 141632, [161 =< 1087.2065s40, 2 10.2,459.1076.00b6.00,00.0071.0000.00D40.0796.0144.0081.0860.19643.0571.0022.0417.0383.0035.1037 0110 3.14150493,[ = 1150.916382, 2=11.0015.00062.210 1. (I0.1000.0003.) 00.0. 0001.0)6 9~3.0019.0001.0001.0007.0900.3884.0002.0000.3262 0110.= 3.463 93= 1252.64C076, 2 = 12.020t6.0t78.0030.0010l.00"37.00C-C.5790).2396..00CC.000.0468.0014.0203.0005.0043.0207.0005.0050 0810 3.141593,9 [RIO = 1324.2386925, 2 13.0004.(,COO.2 i.060.0000C.0002.00 000 0o'.6097.0083.n0c0.0709.0003.0013.0944.0003.0014.1446 uL)C, 3. 141693, [RIO = 1356.861038,. = 14.0181.0072.0000.0003.0010.0000.OUOO.0000.0015.0000.0009.4274.0008.0001.2870.0001.0000.2530 OECC =.1415633, [RIO) = 1448.227066, 2 15 ~0000...00.00.....0005.X0.0CCC. 0000.0000.0000.0052.0469.0000.0001.2824.1910.0002.2828.1909.0001 UELC, =.1415693, FkE0 = 2b54.489990, 2 16,CCOO.C000.0001.(JoQ0C.0)00C.0000.0000.0000.0001.0626.0000.0000.1631.3055.0000.1631.3055.0000 0110 = 0.14I't1 59.2,. FRIO = 29Zi6,605a77,.2-~ - 3=1 -.0002.0001.0000.0000.00CC.0000.0042.0061.0000.0000.1100.0000.1395.3001.0000..1396.3002.0000 DELC = 3.141593, [ _ REQ = 2983.807373, 2=18.0300.089.0000.0000.0000 OOCO0,992.6012.0000..0020.0020.0001 _.Q_1QQ0033.OQOO.0010.0032.0000

UNIVERSITY OF MICHIGAN 3 901 5 03483 82871111111 3 0503483 8287