摘要:

GENERAL DISCUSSIONS OFTHE FARADAY SOCIETYDate190719071910191 1I912191319131913191419141915I9161916191719171917191319181918191319191919192019201920192019211921192119211922192219231923192319231923192419241924192419241925192519261926192719271927SubjectOsmotic PressureHydrates in SolutionThe Constitution of WaterHigh Temperature WorkMagnetic Properties of AlloysColloids and theu ViscosityThe Corrosion of iron and SteelThe Passivity of MetalsOptical Rotary PowerThe Hardening of MetalsThe Transformation of Pure IronMethods and Appliances €or the Attainment of High Temperatures in aRefractory MaterialsTraining and Work of the Chemical EngineerOsmotic PressurePyrometers and PyrometryThe Setting of Cements and PlastersElectrical FurnacesCo-ordination of Scientific PublicationThe Occlusion of Gases by MetalsThe Present Position of the Theory of IonizationThe Examination of Materials by X-RaysThe Microscope : Its Design, Construction and ApplicationsBasic Slags : Their Production and Utilization in AgriculturePhysics and Chemistry of ColloidsElectrodeposition and ElectroplatingCapillarityThe Failure of Metals under Internal and Prolonged StressPhysico-Chemical Problems Relating to the SoilCatalysis with special reference to Newer Theories of Chemical ActionSome Properties of Powders with special reference to Grading byThe Generation and Utilization of ColdAlloys Resistant to CorrosionThe Physical Chemistry of the Photographic ProcessThe Electronic Theory of ValencyElectrode Reactions and EquilibriaAtmospheric Corrosion.First ReportInvestigation on Oppau Ammonium Sulphate-NitrateFluxes and Slags in Metal Melting and WorkingPhysical and Physico-Chemical Problems relating to Textile FibresThe Physical Chemistry of Igneous Rock FormationBase Exchange in SoilsThe Physical Chemistry of Steel-Making ProcessesPhotochemical Reactions in Liquids and GasesExplosive Reactions in Gaseous MediaPhysical Phenomena at Interfaces, with special reference to MolecularAtmospheric Corrosion. Second ReportThe Theory of Strong ElectrolytesCohesion and Related ProblemsLaboratoryElutriationOrientationVolumeTrans. 33678999101011121213131314141414151516161616171717171813191919191920202020202121222223232GENERAL DISCUSSIONS OF THE FARADAY SOCIETYDate1928i 9291929192919301930193119321932193319331934193419351935193619361937193713981938193919391940194119411942194319441945194519461946194719471947194719481948194919491 949195019501950195019511951195219521952195319531954SubjectHomogeneous CatalysisCrystal Structure and Chemical ConstitutionAtmospheric Corroslon of Metals.Molecular Spectra and Molecular StructureOpucal Rotatory PowerColloid Science.Apphed to BiologyPhotochemcal ProcessesThe Adsorption of Gases by SobdsThe Colloid Aspects of Textile MaterialsLiquid Crystals and Anisotropic Meltst r e e KadicalsDipole MomentsColloidal ElectrolytesThe Structure of Metallic Coatings, Films and SurfacesThe Phenomena of Polymermtion and CondensationDisperse Systems in Gases : Dust, Smoke and FogStructure and Molecular Forces in (a) Pure Liquids, and (6) SolutionsThe Properties and Functions of Membranes, Natural and ArtificialReaction KineticsChemical Reactions Involving SolidsLurmnescenceHydrocarbon ChemistryThe Electrical Double Layer (owing to the outbreak of war the meetingThe Hydrogen BondThe Oil-Water interfaceThe Mechanism and Chemical Kinetics of Organic Reactions in LiquidThe Structure and Reactions of RubberModes of Drug ActionMolecular Weight and Molecular Weight Distribution in High Polymers.(Joint Meeting with the Plastics Group, Society of Chemical Industry)The Application of Infra-red Spectra to Chemical ProblemsOxidationDielectricsSwelling and ShrinkingElectrode ProcessesThe Labile MoleculeSurface Chemistry.(Jointly with the Societk de Chimie Physiquo atThird Reportwas abandoned, but the papers were printed in the Trmsuctions)systemsBordeaux.) Published by Butterworths Scientific Publications, Ltd.volume2425252526262728292930303131323233333434353535343737383940414242 A42 BDisc. 12Colloidal Electrolytes and SolutionsThe Physical Chemistry of Process MetallurgyCrystal Growth 5Chromatographic Analysis 7Heterogeneous Catalysis 8Trans. 43Disc.34Lipo-Proteins 6The Interaction of Water and Porous MaterialsPhysico-chemical Properties and Behaviour of Nuclear Acids Trans. 46Spectroscopy and Molecular Structure and Optical Methods of In-vestigating Cell Structure Disc. 9Electrical Double Layer Trans. 47Hydrocarbons Disc. 10The Size and Shape Factor ia Colloidal SystemsRadiation Chemistry 12111314151617The Physical Chemistry of ProteinsThe Reactivity of Free RadicalsThe Equilibrium Properties of Solutions of Non-ElectrolytesThe Physical Chemistry of Dyeing and TanningThe Studv of Fast Reactions1954 Coagulation and Flocculation 1GBNBRAL DISCUSSIONS OF THE FARADAY SOCIETYDate19551955195619561957f 958195719581959195919601960196119611962196219631963196419641965196519661966196719671968196819691969SdjectMicrowave and Radio-Frequency SpectroscopyPhysical Chemistry of EnzymesMembrane PhenomenaPhysical Chemistry of Processes at High PressuresMolecular Mechanism of Rate Processes in SolidsInteractions in Ionic SolutionsConfigurations and Interactions of Macromolecules and Liquid CrystalsIons of the Transition ElementsEnergy Transfer with special reference to Biological SystemsCrystal Imperfections and the Chemical Reactivity of SolidsOxidation-Reduction Reactions in Ionizing SolventsThe Physical Chemistry of AerosolsRadiation Effects in Inorganic SolidsThe Structure and Properties of Ionic MeltsInelastic Collisions of Atoms and Simple MoleculesHigh Resolution Nuclear Magnetic ResonanceThe Structure of Electronically-Excited Species in the Gas-PhaseFundamental Processes in Radiation ChemistryChemical Reactions in the AtmosphereDislocations in SolidsThe Kinetics of Proton Transfer ProcessesIntermolecular ForcesThe Role of the Adsorbed State in Heterogeneous CatalysisColloid Stability in Aqueous and Non-Aqueous MediaThe Structure and Properties of LiquidsMolecular Dynamics of the Chemical Reactions of GasesElectrode Reactions of Organic CompoundsHomogeneous Catalysis with Special Reference to Hydrogenation andBonding in Metallo-Organic CompoundsMotions in Molecular CrystalsOxidationFor current availability of Discwionvolumes, see back cover.Volume19202122232425262728293031323334353637383940414243444546474

ISSN:0366-9033    

DOI:10.1039/DF969480X001

出版商:RSC    

年代:1969

数据来源: RSC

摘要:

GENERAL INTRODUCTIONBY R. J. ELLIOTTDept. of Theoretical Physics, Oxford University1. INTRODUCTIONThe dividing line between solid state physics and physical chemistry is by no meansclear. Indeed from the theoretical point of view I do not think there is any dividingline at all. Dirac says at the beginning of his book that quantum mechanics providesa basic description for a great deal of physics and the whole of chemistry. He means,I suppose, that it describes all electronic and nuclear motions with energies of theorder of electron volts. In this meeting we are concerned only with the motions of theheavy nuclear particles in crystals. The experimental techniques for studying suchmotions are largely common to all crystals, as indeed are the theoretical techniquesrequired for their description and interpretation.If there is a division betweenphysics and chemistry, molecules are clearly chemistry. It is therefore appropriatethat we should consider the application of these physical ideas and concepts tomolecular crystals. Because I am largely unfamiliar with the literature from thechemical side I shall have to draw my analogies in the main from solid state physics.I only hope that I can give a slightly different angle to those who are already familiarwith the field.2. INTERATOMIC POTENTIALSThe essential approximation required to study these motions is the adiabaticapproximation, or Born-Oppenheimer approximation. This states (i) that the motionof the light mass electrons is so rapid that it may be considered as though the heavymass nuclei are fixed and (ii) that the slow motion of the nuclei may be considered astaking place in the field of the electrons averaged over their position.For a diatomicmolecule therefore the energy of the electron states may be determined as a functionV(R) of the internuclear separation R. The nuclear motion is determined as by thiseffective potential energy. The typical curve of V(R) is shown in fig. 1. It has anattractive well at the equilibrium position R = Ro and is parabolic Y(R)cc(R-R,-J2near that separation. At small R it rises steeply because of the large coulombrepulsion between charged particles at small distances. At large R, Y(R) tends to aconstant, and the force F = -3 Vl3R between the particles tends to zero.The energylevels for such a potential are familiar from the vibrational spectra of small molecules.The motion is harmonic and the energy levels equally spaced at low energies. As theenergy is raised the motion becomes anharmonic. The potential, if expanded inpowers of (R-R,), has other terms of which the cubic and quartic are the mostimportant. The cubic term is responsible for thermal expansion. Both terms causethe effective vibrational frequency to decrease as the average energy increases.In a solid, similar potentials V(R) exist between all pairs of atoms, and the dis-placement of any atom is in principle coupled to all the others. As long as the motionremains harmonic it is possible to deal with these couplings in a simple way, but if the8 GENERAL INTRODUCTIONanharmonicity is important it is much more difficult. The anharmonicity of fig.1 isonly important at high energies but there are some cases of low energy anharmonicity.It is convenient to think of these in terms of the displacement of one atom surroundedby a cage of fixed atoms. This is, of course, an approximation; in the crystal thiscage will also move and be coupled to other atoms.FIG. 1.-Usual form of interatomic potential well. The energy levels become closer together as theenergy increases (from A to B) and become continuous (C).A simple case is that of an atom too small for its cage. The potential well willthen have a flat bottom as in fig. 2. Such a well is known to occur for some impuritiesin alkali halides like Cu+ in NaCl or Li+ in KBr where they substitute for a muchlarger atom and in some metallic systems.l They will also occur in clathrate com-pounds where there are large holes in the lattice.In some cases the well may grow ahump in the middle. If this barrier is larger than the zero point energy of a particleFIG. 2.-Flat-bottomed well showing low energy anharmonicity.in a single well the low energy states will correspond to vibrations in the constituentwelts. There wilt, however, be a tunnelling probability with a characteristic frequencybetween the wells. This frequency will depend on the height and thickness of thebarrier (see fig. 3). At high energies the particle will move as in a single well. Thereare several examples of this type of motion important in solid state physics.The Hatoms in KH2P04 move linearly in a double well between pairs of PO4 groups,2and the ordering of the proton positions at low temperature gives rise to the ferro-electric behaviour. A more complicated behaviour occurs when the atom motioR. J . ELLIOTT 9is in two or three dimensions. Oxygen atoms in Si are believed to take up positionsbetween a Si pair and bond strongly to both atoms. The Si-0-Si is bent andpotential in the plane perpendicularly bisecting the Si pair has a central hump. Ifthis barrier were large the 0 atom would effectively execute angular rotations in theplane and radial vibrations. If the barrier is low the whole motion will be like thatof a planar oscillator-in fact, the motion is close to the latter.3 Sometimes, smallions in a cage have a number of equivalent wells.For Li+ in KCl there are six inthe (100) directions, and the ion tunnels between them at low freq~ency.~FIG. 3.-Double well-low-lying levels are vibrations A in separate wells with a low tunnellingfrequency B. Upper levels are vibrations in large well C .3. HARMONIC APPROXIMATIONAs long as all the interparticle potential energies may be regarded as harmonic it ispossible to give a complete theoretical description of the sy~tem.~ Let the crystalhave N unit cells centred at R(Z) with s atoms per cell. Let ua(Z) be one of the 3scomponents of displacement of one of these atoms along a Cartesian axis in the Zthcell. The potential energy is a general quadratic function of theseC C~aa(z,z’>U.(l)uBII‘>.al p l ‘In a perfect lattice the force constants @ depend only on the relative position R(1)-R(Z’) and not on the actual value of 1.The problem of coupled oscillators is familiarin classical mechanics. It may be separated into independent oscillations of thenormal modes. Because of the translational symmetry of the lattice these must havethe same amplitude in each unit cell. They therefore have only a wave-like variationof phase like exp [ik . R(Z)] between cells. The values of k are restricted so that thewavelength is always greater than the unit cell dimensions. In fact, they must lie inthe unit cell of the reciprocal lattice, the Brillouin Zone. If boundary conditions areapplied to make the crystal finite we find that there are just N possible k-valuesuniformly spread in this region.The characteristic frequencies of the modes aredetermined for each k from the determinant.whereI Maa2Sap-QrrjLk) I = 0QUa(k) = EQaa(Z,l’) exp [ik R(1) - R(Z’)].I ’The relative atomic motions in each cell can be derived from the eigenvectors. Thereare 3s modes for each k, and hence 3s continuous distributions of o(k) across the zone10 GENERAL INTRODUCTIONThe total number of modes 3sN corresponds to the number of degrees of freedom.Of these 3s branches it can be shown that three are acoustic where c o ( k ) ~ k at small k.As k-+O these modes correspond to uniform translation of the whole lattice. Theother 3(s- 1) branches are optic and w(k)+const as k+O.Phonons, single quanta of these vibrations, can be excited or destroyed by externalstimuli.Inelastic neutron scattering is by far the most powerful method sinceexcitations of any k may be excited or destroyed and rZw(k) measured directly as afunction of k. The intensity of such scattering also reflects the relative atomicamplitudes in the vibration. If the neutron scattering is incoherent because of arandom distribution of isotopes or nuclear spins the lack of interference betweenscattering off different sites means that the k values cannot be determined. But allquanta of a given frequency will contribute to the scattering and this will be relatedto the density of modes per unit frequency range. Protons are strong incoherentscatterers and hence in hydrogenous materials the density of modes of H motion ismost easily measured.Optical experiments give rather less information since the exciting radiationtypically has wave vectors which are very small on the scale of allowed values of k.They allow measurement of k = 0 modes both in absorption and scattering (Ramaneffect).Selection rules apply so that not all optic modes can be observed with eithermethod. Two phonon spectra are sometimes observed 6-here the modes must havek and - k so that the total k value created is effectively zero. This too is related tothe density of modes, weighted by a coupling factor.These techniques provide basic information about the phonon spectra of solidswhich can be used to predict thermodynamic properties and the like.An interpreta-tion of the data leads back to @(k) and hence may be used to determine the forceconstants between atoms-for comparison with microscopic theories of V(R).If the anharmonic part of the potential is expanded in powers of the displacement,it gives rise to interactions between the phonons. They have a mean free time due tocollisions which gives a broadening of the w(k) curves-these too can be measured byneutron and optical experiments. The width normally increases with T. In addition,the frequencies change with T as the effective curvature changes. Recently therehave been several attempts to calculate these anharmonic shifts by self-consistentmany body techniques.’ They are most important in rare gas solids (the physicist’smolecular crystal).Here the potential wells are weak van der Waals type. If theatomic mass is low the level spacing and the zero point energy are high. Helium is,of course, the classic case-its low melting point and quantum effects arise in just thisway. Phonon spectra have been observed in solid Hes, Kr and Ne.lo In thelatter, detailed interpretation shows second neighbour interactions some 10 % of thefirst neighbour ones, and first neighbour anisotropic forces some 5 % of the mainvan der Waals contribution.4. MOLECULAR CRYSTALSIn a molecular crystal the constituents are molecules which are strongly bondedtogether and then are relatively weakly coupled to each other by van der Waals forcesor hydrogen bonds. Molecules may also occur in ionic crystals-here the coulombinteraction between ions makes for stronger interionic coupling.A molecule withm atoms would in free space have 3 translational, 3 rotational and (3m - 6) vibrationaldegrees of freedom. (For a linear molecule there are only two rotational degreesand (3m - 5) vibrational.) In the crystal the translational and rotational motions areno longer free but are hindered by the intermolecular forces. If the harmonicapproximation holds they are transformed into vibrations of the relative distancR . J . ELLIOTT 11between the molecules. In a simple crystal with one molecule per unit cell there willbe 3m branches to the phonon spectrum. Three of these will arise from the transla-tional motion and form the low-frequency acoustic modes.Three will arise from therotational motion which will be constrained into a vibrational rocking motion called alibration. The other (3m - 6) will be high frequency modes deriving from the internalvibrations. If a typical internal force constant is m0, and typical intermolecular forceconstant CD,, these optic modes will have frequencies given byMu2 = cDo+cDl(k).The mean value will be similar to that in the isolated molecule wo = ((D0/M)* butthere will be a spread over a narrow bandOn the other hand, the spread in the acoustic band will be larger given by49 - (@pl(k>/M>+.The narrow bands corresponding to intramolecular vibrations may be modifiedin other ways by the environment. A symmetric molecule may have degeneratefrequencies-the bands arising from them will not remain degenerate over all k-values.Indeed, the degeneracy will be removed at k = 0 if the environment has lowersymmetry than the molecule.Those internal vibrations which set up oscillatingelectric dipole moments are of particular interest since they are infra-red active.However, in a dense crystal the coupling of these modes to the electromagnetic fieldis so strong that the Reststrahl phenomenon will occur and the crystal becomereflecting near this frequency. These oscillating dipoles will also have an interactionwith force constant larger than the van der Waals type and of longer range so thatsuch branches may be expected to have a larger dispersion than the others.Of course, it is only an approximation to continue to separate the translational,rotational and intramolecular vibrational motion in a solid.The resulting modes willbe coupled together. If a vibrational and a librational band cross they will repel eachother and be strongly mixed near the cross over. But all the modes will be partlychanged. The low frequency modes will also be modified by the existence of theinternal modes. For even at these low frequencies the intermolecular motions willinduce distortions of the constituent molecules. In ionic and covalent crystals asimilar effect arises from the distortion of the atoms and ions on collision. It hasbeen conveniently described by the " shell model " l 1 where the nucleus and outerelectron shells of the atom are allowed to have relative motion.The effects are mostimportant when these distortions are associated with electric dipole moments sincethey give rise to long-range effective force constants. This suggests that a similareffect-apparently long-range forces-might be observed in the acoustic vibrationsof molecular crystals. The determination of the intermolecular force constants forcomparison with microscopic theories is one of the most important results of experi-ments on the phonon spectra.If a crystal contains p different molecules in a unit cell it will have high-frequencybranches near to the internal vibrational frequencies of each constituent. There willbe 6 p low frequency branches representing the relative motion. Of these, 3,u will betranslational representing relative motion of the molecules as a whole ; 3 of them willbe acoustic and 3(p- 1) optic.If there is more than one identical molecule in a unitcell there will be small splittings of the narrow bands corresponding to their internalvibrations. These may only be susceptible to measurement by high-resolutiontechniques-the Raman effect is probably the best method. It is reminiscent of theDavydov splitting of exciton lines, 12 GENERAL INTRODUCTION5. ANHARMONIC MOTION I N MOLECULAR CRYSTALSThe potential curves appropriate to the internal vibrations of molecules are of thetype shown in fig. 1. The anharmonicity is small at low energies. The same is trueof intermolecular forces of the van der Waals type. One anharmonic effect of someinterest is the coupling of optically active internal vibrations to pairs of acousticphonons.A similar process in the " shell model " referred to in $4 gives a mechanismfor two-phonon optical absorption13 in covalent crystals like Si and Ge. Suchabsorption might therefore be relatively large in molecular crystals and give importantinformation about the density of low-frequency modes.Van der Waals forces are central, i.e., they only depend on the interatomic distanceand not on the direction. They therefore produce only small barriers againstmolecular rotation and for this reason the rotational motions of whole molecules orconstituent radicals are often anharmonic at relatively low energies. The potentialenergy will be a periodic function of angle with period 2n or some submultiple of it.For an isolated rotor the nature of the energy states depends on the relative height ofthe barriers Y to the position of the rotational energy levels.If Y<A2/I, where 1is the level of inertia, all levels will look like those of a free rotor. If on the otherhand V>> A2/I the lowest levels will correspond to rocking vibrations (librations) inthe wells. High levels with E> Y will continue to look like rotational levels.Only H2 molecules in solid hydrogen or as impurities show slightly hinderedrotation at the lowest energies. All other molecules appear to show librations untilthe energy is sufficiently large. H2 has, of course, the smallest I-the first rotationallevel is at 15 meV. Because of the form of the nuclear spin functions transitions inwhich the angular momentum J changes by one unit are essentially forbidden thoughthey can be excited by slow ne~tr0ns.l~ Ortho-hydrogen in states of odd J and para-hydrogen in states of even J act as though they are separate species.At low T, well-defined excitations J = 0+2 and J = 1+3 can be seen optically and J = 0-+1neutrons. Such excitations will have a well-defined k because of the translationalsymmetry of the crystal and a dispersion relation E(k) as the excitation moves frommolecule to molecule. These might be called rotons (except that name is used for arather different excitation in liquid He) after van Kranendonk.15 He and hisco-workers have made a detailed study of the properties of H2 and D2.It may be noted that rotons, unlike phonons will have a strong interaction witheach other since two excitations on the same molecule have an entirely different energy.If T is large enough to produce a large number, the concept of independent excitationswill cease to be useful.The effective E(k) curves will become broad as Tincreases.At high T they may all be regarded approximately as free independent rotators buthere too the rotational levels will be broad. The theoretical treatment of suchanharmonic systems away from T = 0 is difficult and no adequate theory yetexists.The dispersion on E(k) is given by the intermolecular coupling which depends onmolecular orientation-electric quadrupole-quadrupole, anisotropic van der Waalsand valence forces.For other molecules the lowest energy levels in the wells are approximately equallyspaced and at low T the few excitations are non-interacting phonons of the librations.But as the number of these increases and energies reach the non-parabolic part of theenergy well the concept of independent phonons becomes less useful.The effectiveE(k) curves for librations become broad. Because of their mixing with other vibra-tions there will be large anharmonic effects throughout the part of the frequencyspectrum where librations give a large contribution to the density of modes.It is only of the order of 0.5 meV in H2R. J . ELLIOTT 136. LOW T DIRECTIONAL ORDERINGIn the situation where librations exist there are often several equivalent minima inthe potential curve.They correspond to different orientations of the molecule, andthe rocking vibrations take place about these positions of equilibrium. A simpleexample occurs in ammonium halides where NH4 ions have two different orientationsin the surrounding cube of halide ions.16 The forces between the molecules can giverise to a tendency towards ordering-in this case it is effectively a short range octopole-octopole interaction. It is convenient to treat the ordering problem in such a systemby analogy with magnetism. We can associate a pseudospin variable S with eachmolecule so that S, = ++ states correspond to the two orientations. The inter-molecular forces can be writtenA? = CJijSfSgi , jas in the Ising model for magnetism.If there is tunnelling between the two states ata frequency T/A an extra term CrS; must be added to 8. An exactly similar Hamil-tonian has been used to describe the H in a double well in KH2P04.17 There areexcitations in this system corresponding to directional changes in the molecules whichare analogous with spin waves in magnetic materials.Another system which shows directional ordering is ortho-hydrogen. The J = 1states are triply degenerate and correspond to three different directions of the molecularaxis. At very low T the anisotropic forces produce an ordering, and there are excita-tions corresponding to directional changes. *For other crystals of diatomic molecules like N2 the ground state has a particulardirectional arrangement.The excitations are librational. As the temperature israised there are phase changes to other structures which allow greater rotationalfreedom. Methane is another example-it shows greater rotational freedom at highT and a transition to an ordered state at low T. The excitations show considerableanharmonic broadening.i5. CONCLUSIONThe harmonic motions in molecular crystals are well understood theoretically.Optical and neutron diffraction techniques allow them to be studied in detail andinterpreted in terms of force constants. But the anharmonic motion, mainly of therotational modes, has as yet no detailed theoretical treatment. The methods whichhave been used in rare gas solids are not immediately applicable to this case. Moredetailed experimental investigation of these motions is needed if they are to beunderstood.A.J. Seivers, Localised Excitations in Solids, ed. R. F. Wallis (Plenum Press, New York, 1968),p. 27.A. A. Maradudin, Solid State Physics, vol. 18, 19, ed. F. Seitz and D. Turnbull (AcademicPress, New York).J. Dash, D. P. Johnson and W. M. Visscher, Phys. Rev., 1968, 168, 1087.R. E. Peierls and F. Goldstein, to be published.G. E. Bacon and R. S. Pease, Proc. Roy. SOC. A, 1955,230,359.R. Bosomworth and W. Hayes, to be published.P. B. Clayman and A. J. Seivers, Localised Excitations in Solids, ed. R. F. Wallis (Plenum Press,New York, 1968), p. 54.E. Kraetzig, T. Timusk and W. Martienssen, Phys. Stat. Sol., 1965, 10, 709.M. Born and K. Huang, Dynamical Theory of Crystal Lattices (O.U.P.).F.A. Johnson, Proc. Phys. Soc., 1959, 73, 26514 GENERAL 'INTRODUCTIONT. R. Koehler, Pliys. Rev., 1968, 165,942.N. S. Gillis, N. R. Werthamer and T. R. Koehler, Phys. Rev., 1968, 165, 951.L. H. Nosanow, J. H. Hetherington and W. J. Mullen, Phys. Rev., 1967, 154,175.F. P. Lipschultz, V. J. Minkiewicz, T. A. Kitchens, G. Shirane and R Nathans, Phys. Rev.Letters, 1967, 19, 1307.T. 0. Bmun, S. K. Sinha, C. A. Swenson and C. R. Tilford, Neutron Inelastic Scattering(I.A.E.A. Vienna, 1968), vol. I, p. 339.W. B. Daniels, G. Shirane, B. C. Frager, H. Umebayaski and J. A. Leake, Phys. Rev. Letters,1967, 18, 548.lo J. A. Leake, W. B. Daniels, J. Skalyo, B. C. Frager and G. Shirane, Phys. Rev., 1969,181,1251.l 1 W. Cochran, Proc. Roy. SOC. A, 53,1959,260.l 2 R. S. Knox, Solid State Physics, Suppl. 5, ed. F. Seitz and D. Turnbull (Academic Press, Newl3 M. Lax and E. Burstein, Phys. Rev., 1955,91, 39.l4 P. 0. Egelstaff, B. C. Haywood and F. J. Webb, Proc. Phys. Suc., 1967,90,681.l 5 J. H. van Kranendonk, Physica, 1959,25,1080.A. D. B. Woods, W. Cochran and B. N. Brockhouse, Phys. Rev., 1960, 119,980.York, 1963), p. 29.R. J. Elliott and W. M. Hartmann, Proc. Phys. SOC., 1967,90, 671.J. H. van Kranendonk and V. Sears, Can. J. Phys., 1964,42,980 ; 1966,44,313.J. H. van Kranendonk and G. Karl, Rev. Mod. Phys., 1968,40, 531.l6 H. A. Levy and S. W. Petersen, J. Amer. Chern. Soc., 1953,75, 1536.J. Freund, Chem. Phys. Letters, 1968, 1, 551.l7 P. G. de Gennes, Solid State Comm., 1963, 1, 132.R. Brout, K. A. Muller and H. Thomas, Solid State Comm., 1966, 4,507.L. Novakovic, J. Phys. Chem. Solids, 1967,27, 1469.J. C. Raich and R. D. Etters, Phys. Rev., 1968, 168, 425

ISSN:0366-9033    

DOI:10.1039/DF9694800007

出版商:RSC    

年代:1969

数据来源: RSC

摘要:

Vibrations in TeflonBY VICTOR LAGARDE, HENRY PRASK AND SAMUEL TREVINOExplosives Lab., F.R.L., Picatinny Arsenal, Dover, New JerseyReceived 11 th June, 1969A measurement of a part of the longitudinal acoustic mode of Teflon (polytetrafluoroethylene)is presented. The measurement was made utilizing the coherent inelastic scattering of low-energyneutrons. The sample consisted of extruded fibres in which the polymer axis was aligned along thefibre axis. The spectrometer used in the measurements provided a monochromatic neutron lineincident on the sample obtained by Bragg scattering the reactor spectrum from a copper crystal.Time-of-flight techniques were used to energy analyze the scattered beam. The observed phononsagree well with a calculation of Boerio and Koenig.Teff on (polytetrafluroethylene) is a synthetic polymer whose chemical repeat unitis CF2.The carbon atoms form a helix. The structure of the helix is known fromX-ray studies.l There exists a phase transformation at 20°C below which there are13 chemical units and 5 turns in the unit cell and above which there are 15 chemicalunits and 7 turns. The present work is concerned with the properties of the 15/7helix since the measurements were performed at ambient temperature (-25°C).The vibrational properties of Teflon have been previously investigated with infra-redand Raman techniques and an assignment of most of the optically active vibrationshas been made.2 In addition, a covalent force model had been successful in repro-ducing this inf~rmation.~ The vibrations which have been observed are, however,not very sensitive to the conformation.The present measurement of part of thelongitudinal acoustic mode is a first step in a series of experiments designed to test itssensitivity to conformation, and to provide better experimental foundation for forcemodels.EXPERIMENTALThe instrument used for the present measurements is a recently constructed time-of-flightspectrometer which is installed on the M.T.T. reactor (5 MW, heavy water moderator).A schematic of the spectrometer is shown in fig. 1. The reactor spectrum of neutrons isincident on a Cu single crystal from which monochromatic neutrons are obtained by Braggscattering. The range of neutron energies available with the present set-up is from 150 cm-',below which the intensity is too small, to 650 cm-'.The upper limit in neutron energy isdetermined by the Cu d spacing and Bragg angle available. The combination of the twochoppers C1 and C2 provide discrimination against second-order reflection from the crystaland time bursts of neutrons incident on the sample. The first chopper C1 is a straight slitcollomator rotating at 6000 rev./min. It produces neutron pulse of 80 p s full width at halfmaximum every 5 ms. The second chopper C2 is a Fermi chopper rotating at 12 000 rev/minand phased with respect to C1 so that only the desired neutrons are present when it opens.The separation of C1 and C2 is 1.2 m. The width of the neutron pulse incident on the sampleis 20 ps. The flight path between the sample and the detector bank is 5 m long.The detector bank, which consists of 9 separate detectors, was split into 3 banks of 3detectors.The difference in scattering angle to the three banks was 1.5" so that the 3momentum transfers corresponding to these three configurations were very little different.116 MEASUREMENT OF LONGITUDINAL ACOUSTIC MODE OF TEFLONHowever, because of the large slope of the dispersion relation in the measured region, theenergy transfer for annihilating one phonon was quite different. Thus, one run of the time-of-flight provided three phonons.FIG. 1.-A schematic of the spectrometer. C1 and C2 are choppers which when phased properlydiscriminate against second-order neutrons reflected by the monochromator XTAL.The detectorsare He3 filled proportional counters.For a time-of-flight instrument, the relationship between energy transfer and momentumtransfer determines the configuration required for establishing a focusing condition. Thus,in fig. 2, are shown two sets of experimental conditions which illustrate a focused and de-focused measurement. Line 1 corresponds to the focus condition in which the (AE,K) linecrosses the dispersion relation as close to normal as possible and line 2 corresponds to thede-focused condition in which the (AE,K) line crosses the dispersion relation close to parallel.The sample of Teflon was constructed by wrapping an extruded fibre around a steelframe. The resulting " crystal " exhibited a rocking curve of 9" full width at half maximum '1-8 2.0 2.2 2.44/4maxFIG.2.-A schematic of two experimental conditions. Line 1 corresponds to the energy againstmomentum transfer condition for Eo = 400 cm-', C$ = 65" and down scattering. Line 2 correspondsto EO = 300cm-', 4 = 67" and up scattering. The third curve is the dispersion relation underinvestigation. The conditions for line 1 correspond to focusing whereas those for line 2 correspondto defocusedV . LAGARDE, H . PRASK AND S . TREVINO 17as measured by neutron diffraction. This is a large mosaic compared to those encounteredin typical coherent inelastic scattering experiments. However, since the inter-chain forcesare so much weaker than the intra-chain forces, little ambiguity is present in the assignmentof the observed phonon as having a momentum parallel to the chain axis.A small mis-alignment of the chain axis, due to uncertainty in a prior knowledge of ~ ( q ) , did not hinderthe observation of the phonon because of the large mosaic.RESULTSMost of the observed phonons were obtained for q/qmax greater than 2 and in thedown scattering mode, (qmax = n/c, c = 1.3 the distance between carbon atomalong the chain axis). A few phonons were observed with up scattering for q/qmaxless than 2 and were found to be consistent with the down scattering data. Onlyone of the acoustic modes was observable, this being the longitudinal mode, since onlythe c-axis direction was known.L-,iI 1 I I I6 0 70 80 90channelFIG. 3.-The time-of-flight spectrum of a typical phonon observation.The channel numberscorrespond to time increments of a time analyzer. The intensity is in arbitrary units. The conditionfor this observation were q = 2.021 qmax, Eo = 200 cm-l, 4 = 34", lzhw = 25.7 cm-l in the downscattering mode.A typical phonon peak is shown in fig. 3. The results of the measurement areshown in fig. 4. The complete dispersion curve could not be measured with thepresent instrument because of the limitation on incident energy and scattering angle.The remaining portion of the mode is at present being measured using a triple-axisspectrometer.Fig. 3 also shows the results of a calculation performed by Boerio and Koenig ;the agreement is excellent. The model employed a generalized covalent force fieldand was used to fit not only the infra-red and Raman frequencies but also the presentneutron data.A previous calculation had resulted in a dispersion curve which wa18 MEASUREMENT OF LONGITUDINAL ACOUSTIC MODE OF TEFLONlower than the present one by as much as 34 cm-1 at the highest observed phonon.The force field required substantial modification thus proving the value of the neutrondata. A similar measurement of the longitudinal acoustic mode in polyethelenehas been made. In this case also, the calculation based on infra-red and Raman dataalone did not agree with the neutron data. The measurement will be extended to4l4maxFIG. 4 . T h e points correspond to the observed phonon. The solid line is from a calculation byBoerio and K ~ e n i g . ~include the entire dispersion relation ; a measurement of the first longitudinal opticalmode will also be attempted. The character of the vibration as a function of tempera-ture, especially below the phase transition, is also under study. This type of measure-ment should prove very fruitful in other polymers. There are a few polymers whichcan be oriented such that the a and b axis are also known. A measurement on thesesamples of the transverse modes should also be possible.The authors acknowledge helpful discussions with Dr. J. L. Koenig and Mr. F. J.Boerio.C. W. BUM and E. R. Howells, Nature, 1959, 174, 549.C. Y. Liang and S. Krimms, J. Chem. Phys., 1955, 25, 563; J. L. Koenig and F. J. Boerio,J. Chem. Phys., 1969,50,2823.M. J. Hamon, F. J. Boerio and J. L. Koenig, J. Chem. Phys., 1969, 50, 2829.F. J. Boerio and J. L. Koenig, Bull. Amer. Phys. SOC., 1969, 14, 406.L. A. Feldknamp, G. Venkataramen and J. S. King, Neutron Inelastic Scattering (Proc. Symp.,Copenhagen, IAEA, Vienna, 2, 159

ISSN:0366-9033    

DOI:10.1039/DF9694800015

出版商:RSC    

年代:1969

数据来源: RSC

摘要:

Studies of Brillouin and Rayleigh Scattering in MolecularCrystalsBY A. J. HYDE, J. KEVORKIAN AND J. N. SHERWOODDepartment of Pure and Applied Chemistry, University of Strathclyde,Glasgow, C.lReceived 16th June, 1969Measurements of the total intensity, the horizontally and vertically polarized components, andthe Brillouin spectrum of light scattered from two molecular crystals, trimethylacetic acid andsuccinonitrile, have been made over a range of temperature in the plastic phase of these materialsup to and a little beyond the melting point. The spectral and intensity changes are correlatedwith the motions available to the molecules in thecrystal.Some organic molecular solids comprised of highly symmetrical molecules undergoa phase transition to yield a highly plastic phase below the true melting point.Sincethe initial observation of this phenomenon there have been numerous investigationsof the nature of this plastic phase. These investigations, which have included infra-red, nuclear magnetic resonance, dielectric relaxation and heat capacity studies 2-1(for pivalic acid and succinonitrile, see specifically ref. (4)-(13)) have revealed that,in this phase, the molecules possess an unusual degree of rotational disorder ; thephase transition being akin to a rotational melting point. Structural studiesindicate that the molecules are translationally disordered. Recently, attention hasbeen turned to the study of the nature of the plasticity of these solids using plasticflow (creep) and radiotracer diffusion techniques.These studies 14* l5 show that themechanism for flow in plastic solids is similar to that for other solid systems at hightemperatures. They also prove that the translational molecular mobility (self-diffusion), which will be rate-controlling in the flow process, is considerably fasterthan in normal organic solids. In order to explain the unusually high plasticity, it wasnecessary to invoke the presence of disordered regions (highly relaxed vacancies) inthe plastic crystal lattice,l4. l5 and one particular consequence is that the regionswould be large enough to scatter light. Some preliminary measurements using aconventional light-scattering apparatus l6 showed that intensity of light scattered wasindeed high and closer to that expected from a liquid rather than from a solid.Therewas however a change in intensity at the melting transition. These results were indic-ative of a high degree of disorder in the plastic phase. The magnitude of the observedeffect was sufficient to warrant the extension of the experiments to a more detailedinvestigation of the intensity and the spectral distribution of light scattered fromplastic crystals in order to obtain further information on the nature of the plastic phase.The two materials initially chosen for examination, pivalic acid (trimethylaceticacid) and succinonitrile, had been previously s t ~ d i e d . ~ Pivalic acid which exists inin the dimeric form in the solid state is the more typical plastic crystal. It undergoesa transition at 280.1 K to the f.c.c.plastic form with an entropy of transition AS, =25.37 J mol-1 K-l and melts at 309.7 K with an entropy of fusion ASf = 6.52 J mol-l120 BRILLOUIN AND RAYLEIGH SCATTERINGI A 1IK-l . The possibility of monomer-dimer equilibrium may complicate the observedphenomena but the fraction of monomer will be small and such interference isunlikely. Succinonitrile is more complicated in its behaviour. The transition to ab.c.c. plastic phase occurs at 233 K, ASt = 28.54 J mol-l K-l, the solid melting at331.3 K, AS', = 11.20 J mol-1 K-I. In the plastic phase the molecule exists in threerotational isomeric forms (two gauche and one trans), the concentration of the latterbeing about 20 % in the plastic phase." The probable effect of this isomerism is toinduce extrinsic disorder in the plastic phase.lAEXPERIMENTALThe Brillouin scattering apparatus used was conventional and is illustrated in blockdiagram in fig.1. The source used was a Spectra Physics 125 helium-neon laser. Thescattering cell was thermostatted to f0.4"C by an external heated jacket and the temperaturewas measured with a calibrated thermocouple. The scanning Fabry-Perot interferometerwas a commercial piezoelectrically scanned model marketed by Hilger and Watts, and basedLASERCHART RECORDER li\siPbl TUBE FABRY-PER07YXTTERINGCELLFIG. 1.-Block diagram of apparatus.on the design of Cooper and Greig.17 The interferometer plates had 85 % reflectancecoatings. The scattered light passes through the interferometer and is collected by a lenswhich focuses the fringe pattern on to a small aperture behind which is a red sensitive photo-multiplier (P.M.) tube (Mullard Type 56 TVP). The P.M.tube could be cooled to liquidnitrogen temperatures if required. The photomultiplier output was fed to a vibrating reedelectrometer which in its turn fed a Bryans X-Y plotter giving a plot of photomultiplieroutput against the voltage applied to the Fabry-Perot scan.The succinontrile and pivalic acid were purified by distillation and zone-refining.Analysis of the h a 1 material by gas chromatography and by melting-point curves indicatedthat the maximum impurity contents would be: succinonitrile (m.p. 331.3 K) 0.04 %,pivalic acid (m.p. 309.0K) 0.004 %.The crystals were grown in rectangular or circularcross-section scattering cells by the Bridgman technique and appeared visually free of grossflaws and inclusions. The light scattering characteristics of both forms of cell were identicaland for convenience most experiments were carried out using the cylindrical cells.In order to be more sure of the absence of inhomogeneities, the total scattered intensitywas measured at 25°C for both materials at various angles to the incident beam and at twowavelengths (435.8 and 546.1 nm) in a SOFICA light scattering apparatus using verticallypolarized incident lightA . J . HYDE, J . KEVORKIAN A N D J . N. SHERWOOD 21RESULTSThe total scattered intensity and its vertically and horizontally polarized com-ponents were measured, for the two materials studied, at a series of temperatures fromabout 20-30 K below, to a few degrees above, the melting point.Brillouin spectra were measured for pivalic acid at several temperatures between20 and 54°C and for succinonitrile at room temperature and in the melt.Measure-ments were also made of the horizontally and vertically polarized Rayleigh-Brillouinspectra for both materials at room temperature and in the melt. The above measure-ments were all carried out using the laser as light source.The experiments using the SOFICA photogonio-diffusometer were made to givean idea of the size of any inhomogeneities. After correction for the volume of solidviewed, the scattered intensity was found to be constant to within 2 % at anglesbetween 45 and 135" to the incident beam for crystals which had not been exposed to afocused laser beam.Crystals which had had laser radiation focused in them exhi-bited an increase in total scatter accompanied by excess forward scatter (roughly 10 %dissymmetry). This increase would appear to have been caused by local melting ofthe crystal in the focused beam followed by refreezing and introduction of inhomo-geneities. Heating effects of laser beams have been noticed previously in liquids. *I I I I I I I I I2 0 3 0 40 5 0 6 0temperature "CFIG. 2.Total intensity scattered at 90" to the incident beam against temperature: A, pivalic acid;0, succinonitrile.temp. "CFIG. 3.-Horizontally (Ho) and vertically (V,) polarized components of scattered intensity anddepolarization ratio ( po) against temperature for pivalic acid22 BRILLOUIN A N D RAYLEIGH SCATTERINGThe increased scatter and dissymmetry could be removed completely by regrowingthe crystals and all data reported refer to experiments using an unfocused laser beam.The total scattered light intensity for the two materials as a function of temperatureis shown in fig.2. The behaviour of the horizontal (H,) and vertical (VJ com-ponents and the depolarization ratio p,, are shown in fig. 3 and 4 for the pivalic acidand succinonitrile respectively. (The incident laser beam was vertically polarizedin all cases). Fig. 5 shows a typical Brillouin spectrum of a pivalic acid crystal andfig. 6 the spectrum of the liquid for comparison.1*- L _ _ I ~ ~ 1 ~ ~ ._ . 1 _ ~ _ i _ I ~ _ _ _ _2 0 3 0 4 0 5 0 6 0T"CFIG. 4.-As fig. 2 but for succinonitrile.FIG. 5.-Brillouin spectrum for crystalline pivalic acid.I 3.4 CH,+3.4 GHz iFIG. 6.-Brillouin spectrum for liquid pivalic acidA. J . HYDE, J . KEVORKIAN AND J . N . SHERWOOD 23From each Brillouin spectrum the hypersonic sound velocity was found from theBrillouin peak splittings. The Landau-Placzek (L.P.) ratio,lg defined as the ratio ofthe intensity of the central peak to that of the two Brillouin peaks, was evaluated usingspectral peak heights since the estimation of the areas under the peaks was sometimesdifficult because of peak overlap. In the cases where peak areas could be determined,the agreement with the peak height ratio was good.The values of sound velocitiesand L.P. ratios are shown in table 1.TABLE 1PIVALIC ACIDtemp. "C nD AV = vs cm-1 velocity ms- 1 L.P. ratios s~gfi:23 1.4 0.212 2033 2.1 6.3630.8 1.4 0.207 1985 2.0 6.2133.9 1.4 0.195 1870 2.0 5.8538.5 1.393 0.1 10 1060 0.67 3.3053.8 1.391 0.117 1129 0.57 3.51SUCCINONITRTLE24.3 1.438 0.252 2353 2.4 7.5661.8 1.417 0.153 1450 0.8 5.59Fig. 6 shows for pivalic acid a total Brillouin spectrum, the horizontal and verticalcomponents of the spectrum and the measured background level. The unusualfeature is the peak in the horizontally polarized component. This peak disappearscompletely in the melt.DISCUSSIONOur main purpose was to try and investigate the formation of defects in the plasticcrystals as the melting point is approached, but the changes which occur on meltingare also of some interest.The main features observed on melting are : (i) decreaseof the hypersonic sound velocity to about half its value for the crystal, (ii) an approxi-mate halving of po and (iii) a decrease of the L.P. ratio.The first observation accords with the few measurements recorded 2o on soundvelocities in organic solids and their melts. Values of p, and the L.P. ratio whilstplentiful for organic liquids 21* 22 do not seem to have been measured for organicsolids and so comparison with other materials is not possible. The values of p, andthe L.P. ratio for the liquid succinonitrile and pivalic acid appear quite normal andthe fall in pv on melting is due to the increase in V,, since H, remains much the sameas in the solid.This increase in V, and also the fall in the L.P. ratio is largely due tothe increase in size of the Brillouin peaks (which are almost 100 % vertically polarized)on melting. This implies high attenuation in the solid which is in accord with knownmeasurements and is similar to the behaviour of viscous liquids on cooling, as observedby Rank et ~ 1 . ~ ~ and Stoicheff et ~ 1 . ~ ~In the crystalline region the most noticeable observations are (i) the small butdefinite decrease in total scatter commencing ca. 10 K below the melting point (theeffect is much greater in succinonitrile than pivalic acid), (ii) the large value of thedepolarization ratio (p,), (iii) the large value of the L.P.ratio, and (iv) the presence ofthe horizontally polarized component of the central peak.The dip in total intensity just before the melting point is difficult to explain. Ananomaly in the proton spin-lattice relaxation time was observed in succinonitrile b24 BRILLOUIN AND RAYLEIGH SCATTERINGPowles, Begum and Norris l 3 and attributed by them to defect diffusion in the sametemperature range.The depolarization ratio is large, being more comparable in magnitude to that fora liquid than for a solid. This presumably reflects the considerable rotationalmotion in these two plastic crystals.The L.P. ratio is more than an order of magnitude larger for the two plasticcrystals than for ordinary hard crystals (e.g., 0.08 for quartz) 25 and is even severaltimes that for ordinary liquids (their own melts included).The values are inter-mediate between those for amorphous transparent solids (e.g., fused silica and glass)and normal crystals. High values of the L.P. ratio have also been reported for someliquids, e.g., ethylene glycol and glycerol when they are cooled to temperatureswhere the bulk viscosity is extremely high, and are attributed to the formation of" ordered regions ".24 The high values of the L.P. ratio in amorphous solids arenormally regarded as arising from " frozen-in " microscopic inhomogeneities. Inthe present case since the materials are predominantly crystalline as evidenced byX-ray 2 6 * 27 and diffusion studies 149 l 5 it would seem more reasonable to attribute thescattering to the existence of small disordered regions within the host lattice.Thescattering experiments at the two shorter wavelengths indicate that the maximumdimension of any defects cannot be greater than about 15nm and may be the" molten " groups of ca. 20 molecules which had been suggested previously 14* l 5 toaccount for difision behaviour and which it was one of the purposes of the presentwork to try and locate.Of particular interest is the spectrum shown in fig. 7. This indicates that theBrillouin lines are totally polarized vertically but that the central line has an appreciable(ca. 10 %) horizontal component. Depolarization values for the central component--L----A-B _ _ _ _ _ - - -_frequencypolarized components, total intensity ( T ) and background (B).FIG. 7.Brillouin spectrum for pivalic acid at 25°C showing, horizontally (Hv) and vertically (V,)as high as this have previously been observed for polar liquids, such as nitrobenzeneby Rank and co-workers.28 On melting our materials, the depolarjzation of thecentral peak disappeared.The observed peak appears to be similar to that observedin glycerol by Stoicheff et aZ.,24 which arises from the coupling of rotational trans-lational motions, and that observed by Rank et al. in nitrobenzene and other liquids.These initial experiments, in confirmation of the work of others, indicate that theplastic phase of these materials is a partially disordered state in which there is conA . J . HYDE, J . KEVORKIAN A N D J .N. SHERWOOD 25Elucidation of more details siderable motion, both rotational and translational.will require further more accurate experiments.We thank the S.R.C. for the award of a grant which made the investigationpossible.J. Timmermans, Bull. SOC. Chim. Belg., 1935,44,17; J. Chirn. Phys., 1938, 35,331.L. A. K. Staveley, Ann. Rev. Phys. Chem., 1962, 13,351.J. G. Aston, Physics and Chemistry of the Organic Solid State, ed., D. Fox, M. M. Labes andA. Weissberger (Wiley-Interscience, N.Y., 1963), vol. I.S. Kondo and T. Oda, Bull. Chem. SOC. Japan, 1954,27,567.H. Suga, M. Sugisaki and S. Seki, Mol. Cryst., 1966,1,377.C. Clemett and M . Davies, J. Chem. Phys., 1960,32,316. ’ D. E. Williams and C. P. Smyth, J. Amer. Chern. Soc., 1962, 84,1808.* G.J. Janz and W. E. Fitzgerald, J. Chem. Phys., 1955,23, 1973.W. E. Fitzgerald and G. J. Jam, J . Mol. Spectr., 1957, 1,49.C. A. Wulff and E. F. Westrum, J. Phys. Chem. 1963, 67,2376.lo T. Fujiyama, K. Tokumara and T. Shimanouchi, Spectrochim. Acta, 1964, 20,415.l2 L. Petrakis and A. Rao, J. Chem. Phys., 1963,39,1633.l 3 J. G. Powles, A. Begum and M. 0. Norris, Mol. Phys., 1969,17,489.l4 G. M. Hood and J. N. Sherwood, Mol. Crystals, 1966, 1,97.l5 J. N. Sherwood, Proc. Brit. Ceram. SOC., 1967,9,233.l6 H. M. Hawthorne, Ph.D. Thesis (Strathclyde University, 1966).l7 J. Cooper and J. R. Greig, J . Sci. Instr., 1963, 40,433.l * J. P. Gordon, R. C. C. k i t e , S. P. S. Porto and J. R. Whinnery, J . Appl. Phys., 1965, 36,3.2o Landolt Bornstein, Numerical Data and Functional Relationshbs (New Series, 1967). 11.L. D. Landau and G. Placzek, 2. Phys. Sowjet Union, 1934, 5, 172.Molecular Acoustics, pp. 15 and 244.J. Cabannes, La Difusion Moltkulaire de la Lumi2re (Presses Universitaires de France, 1929).5.*’ I. L. Fabelinskii, Molecular Scattering ofLight (Plenum Press, N.Y., 1968).23 D. H. Rank, E. M. Kiess and U. Fink, J . Opt. SOC. Amer., 1966,56,163.24 H. F. P. Knaap, W. S. Gornall and B. P. Stoicheff, Phys. Rev., 1968,166,139.25 S. M. Shapiro, R. W. Gammon, H. Z. Cummins, Appl. Phys. Letters, 1966,9,157.26 C. Finback, Arch. Math. Naturvidenskab. B , 1938, 42,71.27 Y. Namba and T. Oda, Bull. Chem. SOC. Japan, 1952,25,225.28 D. H. Rank, A. Hollinger and D. P. Eastman, J. Opt. SOC. Amer., 1966,56, 1057

ISSN:0366-9033    

DOI:10.1039/DF9694800019

出版商:RSC    

年代:1969

数据来源: RSC

摘要:

Lattice Dynamics and Spectral Line Widths of a-N2*BY 0. SCHNEPP 7 AND A. RON $Department of Chemistry, University of Southern CaliforniaReceived 11 th June, 1969The lattice dynamics of a-Nz has been studied, using a potential of the atom-atom form whichhad previously been calibrated to fit the known properties of the solid. Special features encounteredin the treatment of librational motions are discussed. Interactions between translational andlibrational motions are of great importance in determining the dispersion curves and densities ofstates. A study of two-phonon densities together with the experimental observations of line widthsin the optical spectra of the solid leads to the conclusion that there is a considerable difference in theanharmonicities of the two infra-red active translational modes.It is also concluded that the libra-tional modes are much more anharmonic than the translational modes.The observable parameters in the far infra-red and Raman spectra of solid a-N,can be well accounted for in terms of an intermolecular potential model. Thespectroscopic observations included large variations 2* in line widths, which remainedunexplained. In view of these results, it seemed of value to widen the scope of theinvestigations into the lattice vibrations of a-N, and isomorphous molecular solidscontaining simple, linear molecules. Examples of such are C 0 2 and C2N2 whichhave been investigated and discussed as far as their optical properties are con-~ e r n e d . ~ - ~Cochran and Pawley and Pawley have carried out lattice dynamical analyses ofhexamethylenetetramine and of naphthalene and anthracene.These papers containvaluable information concerning the treatment of molecular crystals throughout theBrillouin zone. These authors have also discussed the applicability of the latticedynamical analyses to diffuse scattering of X-rays and the calculation of Debye-Waller factors for molecular crystals.In the present work, the lattice dynamics of a-N, has been studied with severalaims in mind. The structure treated is common to a number of simple molecularsolids for which intermolecular potential models can be calibrated with fair success byusing the measured properties of the solid, including structure and optical spectra.The special properties of the librational or quasi-rotational degrees of freedom werestudied and the interactions between translational and librational lattice modesthroughout the Brillouin zone were investigated.It was of particular interest tostudy the effects of these interactions on the density of states and the densities of two-phonon states. The latter are of interest in connection with the line widths observedin optical spectra and the anharmonicity of the intermolecular potential.* Work at U.S.C. supported by a grant from the National Science Foundation.t Dept. of Chemical Physics, Weizmann Institute of Science, Rehovot, Israel (where Schnepp$ Dept. of Chemistry, Israel Institute of Technology, Haifa (permanent address of Ron).spent a semester on leave from the University of Southern California).20.SCHNEPP AND A . RON 27CRYSTAL STRUCTUREThe crystal structure of a-N2 has been investigated thor~ughly.~ The structure isnearly Pa3 ( T t ) with four molecules per unit cell occupying face-centred-cubic sites.The axes of the four molecules in the simple cubic unit cell have different (111)orientations. The deviation from this structure consists in a shift of the molecules by0.17 A parallel to their figure axes from the centro-symmetric f.c.c.-sites, the resultingspace group being P213 (T4). In all reported spectroscopic investigations, thisdeviation from Pa3 did not manifest itself and no coincidences between infra-redand Raman lines have been observed. In the present work, the approximate andmore symmetric structure was assumed.In order to facilitate the discussion of the symmetry properties of the crystal, thepositions and orientations of the central molecule (of type 1) and of the twelve nearestneighbours have been listed in table 1.The molecules are designated as type 1,2, 33 [- 1 , l - 13TABLE 1 .-MOLECULAR POSITIONS AND ORIENTATIONStype orientation specific molecule location1 [1111 1 (ooo)2 11,- 1,- 11 21 (GO)22 (4-30)23 (-3-40)24 (-330)31 (04%)32 (O+- 3)33 (0 - 3- 3)34 (0- 33)42 (40-4)43 (-30-5)44 <-3o+>c- 1,- 1911 41 (303)and 4 according to the orientation of the figure axis. The nearest neighbour shell of12 molecules is made up of 3 sets of 4 molecules each, the members of a set being ofthe same orientational type and lying in the same coordinate plane.4POTENTIAL MODELThe parametrized potential used has been calibrated to correlate the equilibriumproperties of the solid (cell parameter and crystal energy) with the spectral observables(frequencies and intensity ratios).It is an atom-atom 6-12 potential consisting of 4terms representing the interactions between the non-bonded atoms of two moleculesand is given in eqn. (1) :4 v = C [(a/Rn)'2-(~IRn)6], (1)ll=lE = 2.607 x 10-14erg; cr = 3.356 x 10-*cmHere E and Q are parameters and the Rn are interatomic distances. It was, however,found that this potential with two parameters could not satisfactorily be fitted toreproduce the librational frequencies and a third parameter, the effective bond lengthor distance between the centres of interaction on the molecule, was varied.Theeffective half bond length (designated h) which gave a satisfactory fit to all observableswas 0.4464 A as compared with the spectroscopic value of 0.547 A. This potentialwith 3 parameters was then found to reproduce the cell parameter (by way of the zerouniform stress condition) and the crystal energy within experimental error and the 28 LATTICE DYNAMICS OF a-Nzlattice mode frequencies (2 translational and 3 librational) within 2 cm-l except forthe lowest librational frequency which deviated by 4 cm-l. Also the intensity ratiosin the infra-red spectrum and Raman spectrum could be reproduced with goodaccuracy by using theories developed previously.l** The parametrization of thepotential was carried out for interactions with up to 12 neighbour shells and also fornearest neighbours only.The differences were only of minor nature, as long as theprocedure was consistent. In the present work the lattice dynamical treatment wascarried out in the nearest neighbour approximation only.The crystal potential is assumed to be represented by the sum over pair interactions :4N 4N@ = - r ) c V,,(R,) n = 1...4L t = lwhere N is the number of unit cells in the crystal and the factor 4 enters since there are4 molecules in the primitive unit cell. The R, are distances between the effectivecentres of interaction. The potential is expanded in powers of the displacements aboutthe equilibrium positions and the harmonic term is retained :where eqn. (3) serves to define Qij(2kJ'k') as is usual in this field.ll'LATTICE DYNAMICSTRANSLATIONAL MOTIONSThe treatment of the translational motions is conventional. It is, however,discussed in some detail in order to facilitate comparison with the later treatment ofthe librational motions.The kinetic energy Ttrans is written asTrans = +mc[ti :( Ik) + z i ;( 1 k) + ti :( Ik)] , (4)1.kwhere I designates the unit cell and k the site.The components of the displacementvector u are ui. Here i takes the designations of the Cartesian displacement com-ponents of the molecular centre of mass x,y,z only. We discuss later the librationalkinetic energy in terms of the librational displacement components ue and u8.The equations of motion are derived using the Lagrangian formalismL = T - @ (5)(6)(7)The equations of motion are of the formmii,(lk) = - C @~j(lk,l'k')uj(l'k'),u(2k) = U(kq) exp i[q .r(2k) - m(q)t]?l'k'jwhere the sum over j includes j = x,y,z,6,4. Assuming a solution of the formwe obtain on substitution into the equations of motion (6) :rnw2(q)Ux(kq) = cPx,(ik,l'k')Uj(k'q) exp iq[r(Z'k') -r(lk)] =l'k'jThe last step serves to define M,(k,k',q). The secular equation is now of the form :I Mij(kk',Q) -mm2(q)6kk'6ij 1 = 0. (90. SCHNEPP AND A . RON 29We now consider the symmetry-dictated relations between the potential derivativesQlj(Zk,Z'k'). We can, in fact, express all of these in terms of the derivatives of the pairpotential V,, of a specific pair of molecules in the nearest neighbour shell approxima-tion.We have chosen as such a pair the central molecule, at position (000) and ofaxis orientation [l 1 I] and the molecule of type 2 located at (++O) and of axis orientation[l - 1 - 11 (see table 1). The symmetry transformations used were the inversion in acentre of symmetry i which can be taken as located at any molecular site, the C2screw diads and the C3 triads. The transformation properties of Cartesian vectorcomponents are linear for these transformations and the general discussion of sym-metry transformations given by Cochran and Pawley' and by Pawley apply.Making use of such transformations, all @lj(Zk,Z'kt) can be expressed in terms of the 6distinct second derivatives of V,, for the selected pair of molecules with respect to theCartesian displacement components of one of these molecules :d2 V,,/au,(s) ~ U , ( S ) , with i,j = x,y,z.LIBRATIONAL MOTIONSTo avoid redundancies, two angular displacement components ueti+ were used here.The angles are the usual spherical coordinates.In this sense the treatment differsfrom that given for C 0 2 by Walmsley and Pople for q = 0. In spherical coordinatesfor constant Y, the square of the displacement element is given byds2 = r2du;+ r2 sin2 Odu:, (10)where 0 is measured from the lattice fixed Z-axis to the molecular figure axis, about thecentre of mass of the molecule. We use0 = Go + ug (1 1)and then obtain from (10) for the kinetic energy of the librational motion :qib = $Iz[ir$(lk)+ sin2 O(lk)ic~(Zk)].lkWe now apply the Lagrangian treatment.The appearance of the factor sin2 O(Zk)as coefficient of $$ causes some problems. To date this factor has been treated asfollows. It is argued that the displacements are small l2 and therefore we can expandthe term sin2 O(Zk) about the equilibrium value and retain the first term only, i.e.,sin2 Oo(Zk). However, the validity of this approximation has not been investigatedfor a specific present case, i.e., low frequencies and consequently large amplitudes.We use the described approximation here and leave investigation of the limits of itsvalidity to subsequent work. The equations of motion obtained from the Lagrangiantreatment for molecule (Zk) are then as follows :Iiie(Ek) = - C ~ ~ j ( l k , l t k ' ) ~ j ( l ' k ' ) ,l'k'jI sin2 8,(lk)ii4(lk) = - @4j(lk,ltkt)tij(ltkt).l'k'jBy a treatment parallel to that described for the translational motions, the secularequation is obtained. It is now found that the factor sin2 Bo appears in the rows ofthe secular determinant designated by u9 and as a result this determinant loses itssymmetric structure.However, this difficulty can be overcome by settingu, = sin Bou4, (14)and then we obtain again a symmetric determinant. Alternatively, the determinan30 LATTICE DYNAMICS OF a-N2can be transformed by multiplying appropriate rows and dividing appropriatecolumns by sin O0.The symmetry relations between the CDiJ(Zk,Z'k'> with i, j = 0,a require attention.It was again possible to express all these derivatives in terms of the derivatives of thepair potential Vst of one specific pair of molecules, chosen as before. However, thetransformation properties of the librational displacement components ue and u, aremore complex.We first note that now terms of the type a2X,/aue ( s ) ~ or a2Xn/du, ( s ) ~ do notvanish as they did for derivatives with respect to u,, etc.Such terms occur in thecomputation of a2Vs,/i7ul ( s ) ~ for i = 0 or a, where Vst = Vst(R,) and X, is a Cartesiancomponent of R,. The non-vanishing of these terms is a consequence of the non-linearity of the dependence of the Xn on the ue,u,. To illustrate this point, a typicalexpression for an X, is given below, where R, is the distance between two atoms (orcentres of interaction) on neighbouring molecules s and t :Xn(St) = a/2 + u,(t) - u,(s) + h sin 0(t) cos $ ( t ) - h sin O(s) cos 4(s).(1 5)Here a is the lattice constant, s and t designate the interacting molecules, h is half theeffective bond length (about 20 % smaller than the real bond length for the presentfitted potential). Also 8 is related to ue by eqn. (1 1).It is the consequence of the non-linearity of the above relations (see, e.g., eqn. (1 5))that the transformations of the ue, u4, u, or O,$ under crystal symmetry operationsmust be carried out to second order in the ue,u, when seeking the transformationproperties of the a)@@ and Q44. The necessary transformations are summarized intable 2. The above conclusions also hold if the angular displacement coordinates areexpressed in terms of the rotations about the coordinate axes, a$.As implied, thisdifficulty is only encountered for diagonal elements of the dynamical matrix, Moo andKm.Altogether 8 independent second derivatives and 2 first derivatives of V12 withrespect to librational displacement components had to be computed and all Qij withi, j = 0,4 could be expressed in terms of these by making use of the symmetry trans-formations discussed above.TRANSLATION-LIBRATIONAL INTERACTIONSFrom eqn. (8) and analogous equations obtained for the librations, elementsappear in the dynamical matrix which are of the form M,e or translational-librationalinteraction terms. These terms contain according to eqn.(8) potential derivativesa),,, etc. Since the ui are antisymmetric to inversion for i = x,y,z and symmetricfor i = 0,o (see table 2), the Mxe turn out to be pure imaginary for a centro-symmetriclattice.'. * They do, however, obey the relation M,e = -Me, and therefore thedynamical matrix is hermitian. The symmetry properties of the @,e, etc., are simplyderived from the transformation properties of the ui since here no non-linear termsappear in the calculations of the second derivatives of YI2,. The interaction termsMxe can be expressed in terms of 12 independent derivatives of Y121 of typea2 Vizl/aux(l)aue(l), or a2 v121/k(l)aue(21), etc.SOLUTION OF THE LATTICE DYNAMICS PROBLEMSince the dynamical matrix is hermitian, all real roots are obtained in accordancewith a well-known theorem. Due to the special symmetry of the matrix obtainedhere, i.e., two square symmetric blocks on the diagonal containing only real terms(the 12 x 12 translational block and the 8 x 8 librational block) and the off-diagona0.SCHNEPP AND A . RON 31TABLE 2. -TRANSFORMATIONS OF LIBRATIONAL DISPLACEMENT COORDINATES UNDER CRYSTALSYMMETRY OPERATIONS(All transformations are carried to second order only).purely imaginary 12 x 8 block of interaction terms, the dynamical matrix can be trans-formed into a symmetric and real matrix of the same dimension (20 x 20) by the simpletransformationiue(1k) = u*(lk),iu,(lk) = ur(lk). (16)The solutions give then real eigenvectors in terms of the ux,u,,,uz,u~,u~ but resubstitu-tion in terms of ug and 21, from (16) shows that the eigenvectors are indeed complexand the physical significance of the complex form is to be found in a phase differenceof 7c/2 between the translational niotioas and the librational motions.7* 32 LATTICE DYNAMICS OF U-N,Since the primitive unit cell is a cube in the present case, the Brillouin zone is alsoa cube of side 27c/a.The positive axes of the zone were divided into 15 equal parts(for q between 0 and n/a) and the dynamical matrix solved at 816 points, having dif-ferent weighting factors, the most general point having weight 48. The total numberof points covered throughout the Brillouin zone was 2.70 x lo4 and at each point wehave 20 frequencies (not necessarily all distinct) corresponding to 20 different eigen-vectors.4+interactions.FIG. 1 .-Dispersion curves for CI-NZ, [ 11 11 direction, in the absence of translational-librational9 3FIG.2.-Dispersion curves for a-N2, [ 11 11 direction, including translational-librational interactions0. SCHNEPP AND A . RON 338 0 [ 1 0 0 1 -9+FIG. 3.-Dispersion curves for a-N2, [ 1001 direction, in the absence of translational-librationalinteractions.80/_---706050nId40Wt >302010n9 3FIG. 4.-Dispersion curves for a-N2, [ 1001 direction, including translation-librational interactions.34 LATTICE DYNAMICS OF a-Nzh 1I EWI >4+FIG. 5.-Dispersion curves for wN2, [l lo], direction, in the absence of translational-librationalinteractions.q -27FIG.6.-Dispersion curvcs for a-Nz, [110] direction, including translational-librational interactions.DISCUSSIONDISPERSION CURVESThe dispersion curves for three directions in the Brillouin zone [ l l l ] , [loo] and[ 1 101, each for two cases-with and without translational-librational interactions-are shown in fig. 1-6. The representations to which the modes belong at q = 0. SCHNEPP A N D A . RON 35(factor group T,) are indicated in the figures. The translations are odd to inversionand belong to representations T,, E, or A,, whereas the librations are even and belongto Tg or E, at the zone centre. In absence of interaction with translations, thelibrations show remarkably weak dispersion, and cluster between 30 and 35 cm-l.On the other hand, translational-librational interactions are strong and cause im-portant changes in the structure of the dispersion curves and densities of states.For the [lll] direction, the group of the q-vector is isomorphic to C3 and as aresult only 2 representations occur, one doubly degenerate (here labelled E ) and theother non-degenerate (here labelled A).The double degeneracy is caused by time-reversal symmetry. In the absence of interactions, the acoustic branches have largedispersion and cut across the librations. However, the non-degenerate acousticbranch A interacts with the corresponding optic branch arising from A, at q = 0.When the interactions are included, these cause an effective lowering of the acousticbranches which now interact strongly with the librations.This results also in aconsiderable decrease of the dispersion of these branches. Also, at the same time,the previously pure librational branches are split up and cover a wider range offrequencies. This change is strongly reflected in the densities of states shown infig. 8, 9.Similar remarks to the above apply to the dispersion curves for the [loo] direction.However, since the [loo] axis in Pa3 (T') is only a two-fold axis, all degeneracy isremoved here and the group of the q-vector is isomorphic to C2". As a result, 4different representations occur and interaction between branches is thereby keptwithin limits although the number of branches is large (20 in principle). For theTI- , - - il-i~ , - - i - - ,- , ~ - - - ~ -0-275-- Iij (cm-l)FIG.7.-Density of states for m-Nz in the absence of translational-librational interactions.[ 1101 direction, only a glide mirror plane is preserved and the group of the q-vectoris isomorphic to C,. Now, only two representations occur and the large number ofbranches which arise in the absence of degeneracy interact and give rise to a verycomplex pattern (fig. 6).DENSITY OF STATESThe density of state curves presented in fig. 7 and 8 were obtained by use of aFig. 7 shows that programme adopted from the work of Gilat and Raubenheimer.l36 LATTICE DYNAMICS OF U-Nzthe density of states in the absence of interactions is highly peaked in the regionof the centre of gravity of the librational modes (near 32 cm-l). With the inclusionof interactions, the density of states is spread to lower frequencies (fig.8) in accor-dance with our discussion for the dispersion curves. As already stated, the originallyacoustic branches have considerably lower dispersion. It is also evident from fig.8 that now the curve splits up into two peaks one between 25-28 and the other at32 cm-l. This change also reflects the effect of the interactions and can be correlatedwith the structure of the dispersion curves. At higher frequencies, i.e., in the 60-70cm-l range the interactions have little effect, as expected.0.1 00083 u 3 0.06v,ruh2 0.04u .Ma"002IFIG. 8.-Density of states for a-Nz including translational-librational interactions.TRANSLATIONAL-LIBRATIONAL INTERACTIONIt is concluded in view of the above discussion of the dispersion curves and thedensity of states curves that the interactions between librations and translations arestrong in the potential model used here.This result is considered to be generallyvalid in molecular solids.' A study of the eigenvectors of the normal modes supportsthis conclusion. For example, the lower acoustic branch is principally translationalin character near the zone centre, but the degree of admixture of librational characterrises rapidly and half-way to the zone edge it is typically 30 % librational in character.The degree of mixing increases toward the zone edge as the branches move nearer.In the frequency region 35-45 cm-l there is a preponderance of librational characterwith appreciable translational admixture.At higher frequency, the modes are againpredominantly translational in character.LINE WIDTHSIt was one of the immediate aims of the present work to investigate the mechanismsresponsible for line widths as observed in the far infra-red and Raman spectra of thelattice modes of a-N,. The discussion will here be limited to the standard model l3*l4according to which the line width is determined by the relaxation of an optic phononat q = 0 to two other phonons subject to appropriate selection rules. The perturba-tion hamiltonian which gives rise to the coupling between the states is the anharmoni-city correction. The line width or inverse of the relaxation time is then obtained i0. SCHNEPP AND A . RON 37the form of a transition probability which consists of two factors, (i) the matrixelement of the anharmonic perturbation between the optic phonon (q = 0) and thetwo-phonon state, and (ii) the density of states.In the present work, the density oftwo-phonon states subject to conservation of energy and q-vector were calculated forthe experimentally observed optic phonons in the infra-red and Raman spectra. Thesevalues in units of 1 /cm-l per point in q-space are listed in table 3 for the cases includingand excluding interaction between translations and librations. The densities ofstates were found to be insensitive to varying the energy interval between 2 and6 cm-l. In table 3 we also list the experimentally observed line widths, those forthe lines at 31.5 and 35.8 cm-l having been measured in the Raman spectrum andthose at 48.8 and 70.0 cm-l in the far infra-red spectrum.TABLE 3.-TWO PHONON DENSITIES OF STATES IN UNITS OF l/Cm-l PER POINT IN q SPACE.THE DENSITIES ARE LISTED FOR TWO MODELS : IN ABSENCE AND PRESENCE OF TRANSLATIONAL-LIBRATIONAL INTERACTIONS.optic phonon 2-phonon dens. 2-phonon dens.F (cm-1) expt.expt. line width cm-1 no interactions includinginteractions31.5 (R) 1.4 1.3 x 3.4x35.8 (R) 1.4 3 . 0 ~ 9 . 6 ~48.8 (i.-r.) 0.3 0.28 1.670.0 (L-r.) 6 3.8 3 .OThe experimental infra-red line width ratio is about 20. The two-phonondensities are in the ratio 13.6 when interactions between translational and librationalmotions are not included, but the ratio is less than 2 when interactions are included inthe calculation. 'It is clear that the interactions have a profound effect on the results.The density of states appropriate for the relaxation of the 70.0 cm-1 mode is decreasedby about 20 % on inclusion of interactions, and this change is understandable interms of a spreading of the densities of one-phonon states as already discussed.However, the increase of the density of states appropriate for relaxation of the48.8 cm-l state by a factor of more than 5 is surprising. The cause for this change isin the large decrease in dispersion of the formerly pure acoustic branches, whichcauses a steep increase of the density of states at low frequencies and a correspondingincrease in the density of two-phonon states with total energy near 50 cm-l.Thepresent results show then that the large line width ratio of the infra-red lines cannotsimply be accounted for in terms of a correspondingly large ratio of the density oftwo-phonon states. It is then concluded that the higher frequency mode is con-siderably more anharmonic, a trend which is to be generally expected in solids likea-N2, but the quantitative difference is of interest. However, from the above, amore quantitative discussion of line width ratios in a-N, must await a more detailedtreatment, which includes a careful investigation of the matrix elements of theanharmonic perturbation.For the Raman-active lines, the two-phonon densities of states are very low, asexpected for such low frequencies. The observed line widths show, therefore, thatthe librational modes which give rise to these lines, are considerably more anharmonicthan the translational modes, by a factor larger than 10. This result is of importancesince it underlines the necessity to investigate the applicability of the harmonicapproximation to low-frequency librations in molecular solids.We acknowledge much valuable help received in the course of this work fromDr. M. Brit. We also acknowledge valuable discussions with M. H. L. Pryce,S. Lifson, A. Warshel and N. Cressy and H. Friedmann38 LATTICE DYNAMICS OF G!-N,T. S. Kuan, A. Warshel and 0. Schnepp, J. Chern. Phys., 1970, 52, 3012.R. V. St. Louis and 0. Schnepp, J. Chem. Phys., 1969,50,5177.M. Brit, A. Ron and 0. Schnepp, J. Chem. Phys., 1969,51,1318.S . H. Walmsley and J. A. Pople, Mol. Phys., 1964, 8,345.M. Ito, unpublished. J. E. Cahill, K. L. Trenil, R. E. Miller and G. E. Leroi, J. Ch em. Phys.1967,47,3678.P. M. Richardson and E. R. Nixon, J. Chem. Phys., 1968,49,4276.G. S . Pawley, Phys. Stat. Sol., 1967,20,347.T. H. Jordan, H. W. Smith, W. E. Streib and W. N. Lipscomb, J. Chem. Phys., 1964,41, 756.lo 0. Schnepp, J. Chem. Phys., 1967,46,3983.l1 M. Born and K. Huang, Dynumicd Theory of CrystalLattices, (Oxford Univ. Press, 1954).l2 H. Goldstein, Classical Mechanics, (Addison-Wesley Publ. Co. Inc., Reading, Mass. 1950),l3 J. M. Ziman, Electrons andPhonuns, (Clarendon Press, Oxford, 1960), p. 129l4 D. A. Kleinman, Phys. Rev., 1960,118,118.l5 G. Gilat and L. J. Raubenheimer, Phys. Rev., 1966, 144, 390.' W. Cochran and G. S. Pawley, Proc. Roy. SOC. A, 1964,280,l.p. 320

ISSN:0366-9033    

DOI:10.1039/DF9694800026

出版商:RSC    

年代:1969

数据来源: RSC

摘要:

Statistical Dynamics of a Classical Hindered RotatorBY B. LASSIER AND C . BROTLaboratoire de Chimie-Physique, 91 Orsay, FranceReceived 16th June, 1969The motion of a classical linear rotator hindered by a two-well potential and submitted to randomtorque-impulses is simulated numerically. Various time autocorrelation functions are extracted, andsome corresponding spectral properties are calculated.The previous treatments of the dynamics of the multi-well model for orientationalfreedom suffer from one or more of the following restrictions : (i) The depth of thewell is supposed to be greater than kT; (ii) only the long-time behaviour of thesystem is considered; (iii) the ratio of the rate of changes of wells to the rate ofcrossing of the barrier crest is assumed to be a constant close to unity.Without theserestrictions, one is faced with the general problem of the statistical dynamics of ahindered rotator in contact with a thermal bath. The aim of this paper is to treatthis problem. Since an exact general solution seems a formidable task, a numericaltreatment has been used.The nature of the contact with the thermal bath must first be specified. Thiscontact is achieved through fluctuating torques acting on the rotator. We representthese fluctuations by impulses which are random in their strength and their time ofoccurrence, i.e., by instantaneous impacts from molecules of the medium. It isessential to assume that these impacts produce no discontinuous change in theorientation of the rotator.They only alter its angular velocity in a random manner,as described below.We choose a classical linear rotator along a unit vector u(t), with a moment ofinertia I, and we characterize the randomness of the torque impulses as follows.(i) The impulses take place at random times governed by a Poisson law of charac-teristic time zi (mean time between " collisions ").(ii) The direction of the torque vector is random in the plane perpendicular to therotator. The component of the angular velocity of the rotator which is parallel to thetorque vector is left unchanged by the torque impulse, while its perpendicular com-ponent takes a new value w independent of the previous one.* This random valuew is governed by a Boltzmann law for one angular degree of freedom ; the distributionfunction for w isf(w) = JI/2zkT exp -(Iw2/2kT).The components of the angular velocity along the polar and azimuthal directions arethus= w cos c-(& sin 8 cos r - 8 , sin 5) sin 5, 1; sin 8 = w sin {+(dr sin 8 cos c-8, sin 5) cos c,* These mechanical conditions are equivalent to that of a small hard smooth sphere rigidly boundto the origin, in collision with an identical free sphere : then the torque-impulse is perpendicular tothe line of centres of the spheres, and only the velocities along this line are exchanged.39(140 STATISTICAL DYNAMICS OF HINDERED ROTATORwhere the indices i and f refer respectively to values before and after the torque-impulse, and where 5 is the random angle between the torque vector and the azimuthaldirection d8 = 0 on the unit sphere.The potential which hinders the motion of the rotator (" crystalline field ") hasbeen chosen to be a simple symetrical potential with a double well :W(8) = Vsin2 8.The equations of motion for the rotator areIsin2 84 = J,,21sin2 8)++Id2 + V sin2 8 = E, (3)where J, and E, the angular momentum along OZ and the total energy respectively,are constants of the motion between the impacts.The motion of the rotator has been integrated with a computer using the aboveequations,* the random numbers necessary for determining the times and theparameters of the impacts being generated by standard programmes.The motionhas been recorded for a time long enough for the following autocorrelation functionsto be built by numerical averaging.(1) The vector correlation functions Fvll and FvL which pertain to dipolarabsorption-dispersion in single crystals.Here either Fvll or FVL must be used whenthe radiation is polarized parallel or perpendicular to the polar Oz axis, respectively :Fvl = (sin 6(0) cos 4(0) sin 6(t) cos $ ( t ) }= (sin 8(0) sin #(O) sin O(t) sin +(t)).(2) The vector correlation function relevant to dipolar absorption-dispersion inFv = Fvll + 2FvI = (COS 8(0) cos 8(t) + sin 8(0) sin 8(t) cos [@(O) - $@)I>lovl, = (cos e(o) cos e(t)>,polycrystalline samples= (u(0). u(t)>.Neglecting the internal field corrections, the spectral properties in the microwave-far i.r. region have been calculated from Fv through the Fourier transforms :Go err ClClZ= C$ Eo-n2 co(eo-n2)F,(t) sin cot d t , - = F,(t) cos a t dt.(3) The angular momentum correlation function relevant for spin rotationinteractionF J = (J(0) J(t)>/<J2).(4) The tensor correlation function relevant for Raman scattering with radiationpolarized along Oz, or for dipole-dipole interaction in n.m.r.(Ho along Oz)The following facts support the realistic character of our simulating process.whatever Y and Ti.for V = 1, 3 and 5 kT respectively. The corresponding theoretical values,F= = 4-33 cos2 e(o) - i)(3 cos2 e(t) - 1)).(i) The mean kinetic energy )1(d2+4" sin2 O} was observed to be (1 k0.02) kT(ii) The mean potential energy has been found to be 0.565, 1.10 and 1.14 kTV r ' 2 exp (- V sin2 O/kT) sin3 8 d8 exp (- V sin2 8/kT) sin 8 do.0 is:"*Although precautions must be taken with eqn.(3) when 0 reverses its sign, these equationshave been found to be more tractable than the equivalent second-order differential systemB . LASSIER AND C . BROT 41are 0.57, 1.12 and 1.17 kT respectively. For still higher potentials, the potentialenergy decreases again towards unity due to the near-harmonic character of the bottomof the wells.(iii) In all cases the computed vector autocorrelation function exhibits the correct“ inertial ” (or “ dynamical coherent ”) behaviour at short times1 :FV(t) = 1 -(kT/I)t2 +. . .,with the consequence that the real dielectric constant E’ reaches its high frequencylimit n2 through lower values and that the absorption coefficient per unit length CItends to zero at very high frequencies.I.ZERO POTENTIAL CASE(^^^. 1-3)For V = 0, J does not change spontaneously, but only because of the impacts.Since these are governed by a Poisson law, FJ is an exponential with a characteristictime zJ = 2.ti, because one impact randomizes one out of two velocity components.FIG. 1 .-Zero potential case. Vector correlation functions; time in units (I/kT)$; Points : numericalresults; two upper full lines, formula (4); lower full line, Kummer function.For very rare impacts, Fv approaches the free rotation correlation f ~ n c t i o n , ~- F l;+i;---- - t2 T’l] (Kummer function). c 2 (- l)k(t2kT/l)k1 . 3 . 5...(2k-l) =For more numerous impacts our Fv are identical with the correlation functionscomputed by Gordon for an analogous model (Gordon’s ‘‘ J diffusion model ”,with his z twice our TJ.For very frequent impacts [compared with ,/(kT/I)], Fv has been checked to beessentially exponential : Fv --yo exp (- t/zD).This corresponds to the Debye pictureof dielectric relaxation in compressed gases or non-quasi-crystalline liquids (rotationaldiffusion by small steps). However, due to the correct departure of Fv from expo-nential behaviour at very short times, the transparency of the medium is recovered inthe region co = 1 /z, = 1 /(22,) (see fig. 3), in agreement with the Rocard modificationof the Debye treatment.Shimizu has predicted that in processes of this kind, where the angular velocity israpidly randomized, one should have zDzJ = 1/(2kT).Hence zD = I/(4kTz,) in ou42 STATISTICAL DYNAMICS OF HINDERED ROTATORmodel. For example, in units ,/(I/RT), 7f = 0.01767 should yield zD = 14.14.Our numerical result is 14.1.It is moreover possible to use an analytic form which approximates closely toFv in these cases : the interval between impacts being short, it is permissible to replace0 I(&'- n2)/(E* - n2)FIG. 2.-Zero potential case. Complex dielectric constant plots; from bottom to top Ti = 001767,0.1, 0.2, 0.875 in units (Z/kT)*. The maxima of E" are reached for the angular frequencies: 0.07,0.4, 0.855 and 1.6 (kT/I)* respectively.ac.1 I 100FIG. 3.-&ro potential case. Dipolar absorption coefficient per unit length in units ( ~ ~ - n ~ ) (kT/!)+nc against the angular frequency in units (kT/Z)*.From bottom to top at maximum absorptionTi = 0.01767, 0.1, 0.2,0.875 and 00 in units (I/kT)*.the various angular velocities by their r.m.s. value oo = ,/(2kT/I). This amounts toreplacing the rotator by a spatial oscillator of proper frequency coo. With l / z as aparameter proportional to the frequency of the impacts, the correlation function forthe position of such an oscillator is exactly :'**V(t) = exp (- t/z)[cosh J1/z2-mit+(l - T ~ W ~ ) - * sinh ,/I/T~-W;~] (4B . LASSIER AND C . BROT 43withThe correlation function (C.F.) for the velocity of the oscillator is 'W(t) = exp (-t/z)[cosh J1/z2--W~t-(1-z2co;)-* sinh J1/z2--w;f] (6)If 1 coo, both C.F. are essentially exponential (the first terms in the forms (4') and(6') dominate, since z1 x 2/(w3)>>z/2 z2). Identifying V(t) with Fv(t), z1 with zD,and the characteristic time z2 of the velocity correlation function of the oscillatorwith the correlation time zJ of the angular momentum of the rotator, we obtainZ ~ T ~ = coo-2 = 1/(2kT). The second term in (4') expresses the initial non-exponentialbehaviour caused by inertial effects.It is worth noting that (4') yields exactly thesame spectroscopic formula as the models derived through the macroscopic conceptof viscosity.g-11. DEEP WELLSVery deep wells are not tractable by our method because changes of wells do notoccur often enough. Our results are for V in the range 3-7 kT and are thoughtsufficient to illustrate the phenomena involved. These include for the dipole orienta-tion (fig.4) a long-time behaviour which is essentially exponential (activated changesof wells) and a short-time behaviour which reflects the more or less damped librationalmotion in the wells.< 0 . 5 -0.2t I 1 I 1 I0 5 10 15 2 0tFIG. 4.--V = 3 kT. Vector correlation function, times in units (I/kT)*.(a) LONG-TIME BEHAVIOUR OF F,(t)It has been pointed out recently lo that the orientational relaxation rate-i.e.,l/zD, the reciprocal characteristic time of the exponential part of &(t)-is not only afunction of the height and shape of the barrier, as usually assumed,ll* l2 but dependsalso on the randomization rate of the angular velocity, here (2Q-l. It is true that p ,the number of crossings of the crest line of the barrier per unit time can be theoreticallyderived through equipartition arguments,' independently of the randomizationrate of the angular velocity.However, each crest crossing does not necessarily leadto a change of wells which is complete enough and has sufficient duration to influenc44 STATISTICAL DYNAMICS OF HINDERED ROTATORthe long-time behaviour of the orientational correlation function : on the one hand,when zi is very long, one crest crossing may be followed by several others in a stage ofcontinuous (non-uniform) rotation; on the other hand, when zi is very short, themotion is of a diffusive nature and several back-and-forward crossings of the crestline may occur before the rotator diffuses into any of the wells.Numerical samples ofboth cases have been observed and are reproduced in fig. 5. For intermediate valuesof zi the relaxation rate l/zD is a maximum and tends to be of the order of p , as hasbeen estjmated,1° and not equal to 2p, as with a transmission coefficient unity.l'L- - 10 qz- - - - - 4FIG. 5.--V = 3 kT. Numerical samples of multiple crest-crossing. Times in units (I/kT)*.(a) full line, Ti = 0.01767; (b) dashed line, T i = 5.Table I summarizes our results for the long-time behaviour of the dipole motion.It gives the observed dipolar relaxation rates l/z,,, together with the observed meanfrequencies of crest crossings, p . For a given potential depth, p is close to itstheoretical value, which is lo(kT/2d)* exp (- V/kT)exp (- V sin2 0/kT) siii 0 d0Pth = n/2 0J OThe observed values of l/zD are much smaller than p in both extreme cases of veryshort and very long zi.For rare impacts, we have also recorded the mean rate ofoccurrence of stages during which the total energy keeps a value larger than V. Asexpected, this rate tends to be equal to the dielectric relaxation rate, i.e., is slower thanthe mean crest crossing rate p . The ratio of the latter to the former is indeed equalto the mean number of crest crossings in a cluster such as that reproduced in fig. 5b.(b) SHORT-TIME BEHAVIOUR OF Fv(t)Contrary to the zero potential case, the (long-time) exponential part of &(t)extrapolates at zero time to a value smaller than unity. This is connected with thelibrational freedom in each well (Bauer's " instantaneous polarization ").Thislibrational motion is revealed by more or less damped oscillations of Fdt) between itsinitial value unity and its exponential asymptotic value. If the impacts are sufficientlyrare, these oscillations are clearly distinguishable. Due to the remaining anharmo-nicity of the populated regions of each well, their observed period is slightly larger thanthe harmonic approximation n(21/V)3 obtained by a small angle expansionB . LASSIER AND C. BROT 45TABLE NUMERICAL RESULTS FOR “ DEEP ” WELLS. THE MEAN IMPACT INTERVALS Ti ARETHE BARRIER ENERGY Fe, AND THE LONG TIME DIPOLAR RELAXATION RATE 1 /ZD ARE GIVEN ININDICATED IN UNITS (z/kT)4. THE CREST-CROSSING RATE p , THE RATE OF EXCITATION ABOVEUNITS (kT/Z)*.3 0.0125 9 2.340.1 8.2 7.6(PtheorX 10’ = 9-51 0.2 9.9 8.00.5 9.6 7.83 10.3 2.4 2.35 10.0 1.37 1.255 0.1 2.2 2.5h e o r x 10’ = 2.3) 0.2 2.3 2.60.5 2.4 2.55 2.6 0.4 0.37 0.1 0.41(PtheorX 10’ = 0.44) 0.2 0.500.310.49The total spectra of the rotator (fig.6 and 7) shows, beside the relaxational lowfrequency domain, a second region of absorption which corresponds to this librationalmotion. The deeper the potential wells, the better the separation of the relaxationaldomain and the librational band. The rarer the impacts, the narrower the librationalband, as expected.c’’/(cO - n2)- 0.5 0 0 . 5 1-0FIG. 6.-Complex dielectric constants : full lines V = 3 kT. From top to bottom on the left side(high-frequency) Ti = 5, 0.5, 0.2 and 0.1 (I/kT)*; dashed line V = 7 kT and q = 0-2 (I/kT)&.Thelibrational maxima of E” (left parts) are reached for 2.1; 2.0; 1.6; 1.0 and 3.1 (kT/I)* respectively.The frequencies for the relaxational maxima are given in table 1.Some of the spectra (especially those for short z,) are reminiscent of recent experi-mental data on rotational solid phases.*13 In liquids also the far i.-r. spectra often* However, an eight- or twelve-wells potential would probably be more realistic in many plasticcrystals, as in an approximate model previously proposed. l46 STATISTICAL DYNAMICS OF HINDERED ROTATORexhibit an absorption exceeding the limiting value computed from microwaveresults.15* l6 This behaviour may well be due to a situation corresponding to thepresent model, the “ liquid lattice ” affording the required anisotropic potential, assuggested by Hill. However, even in quasi-crystalline liquids, the anisotropicpotential has a finite lifetime in a given direction, so that the model cannot be morethan a first order approximation.For more exact results, one should resort to many-body molecular dynamics computations.0FIG. 7.-Absorption per unit length (same units as in fig. 3) for V = 3 kT. From top to bottom atmaximum absorption Ti = 5,0.5,0.2 and 0.1 (I/kT)*.0 5f T -f “A-0- I0+20tFIG. S.--Various correlation functions for V = 3 kT. Time in units (l/kT)3; full lines Tj = 5 ;dashed lines Tj = 0.1.To return to our model, overall correlation functions and overall spectra have beenobtained : for example, the recovery * of the transparency of the medium at infinitefrequencies is a necessary consequence of the starting hypothesis.It is only forconvenience that the long- and short-time behaviours have been separated, a dichotomywhich would be meaningless in 3 111.* Contrary to the zero potential case, this recovery takes place in a spectral region which is notsimply related to TD. Hence, the danger of substracting a Debye-Rocard curve from experimentaldata in the present casesB . LASSIER AND C . BROT 47(C) OTHER CORRELATION FUNCTIONS (fig. 8)It is confirmed that the long-time slow decrease of F,, previously described, is dueto changes of wells, since Fvll exhibits this same behaviour, while FvL does not.Theshort time that FvI spends to reach negligible values is in fact the time of establishmentof the polarization that Bauer l1 calls " instantaneous ". For rare impacts, FvLexhibits oscillations at the librational frequency. The initial decrease of FJ is alwaysfast, being due at least to the reversal of the velocities under the influence of thepotential. For very frequent impacts, this decrease is even faster and is not followedby oscillations as in the rare impacts case. FT reaches quickly its finite asymptoticvalue. This is because in our two opposite wells case, FT is not sensitive to the slowprocess of the changes of wells. For rare impacts, FT exhibits a few oscillations attwice the librational frequency, as expected.t73 = 0.1.FIG.9.--Correlation functions for Y = kT. Time in units (I/kT)4. Full line Ti = 5 ; dashed lines0.1 I 10wFIG. 10.-Absorption (same units as in fig. 3) for V = kT. From top to bottom Ti = 5 and 0.1(I/kT)%48 STATISTICAL DYNAMICS OF HINDERED ROTATORIII. SHALLOW WELLS (fig. 9-10)Here there is, to ow knowledge, no previous theoretical work for comparison.For short zt, the process is rotational diffusion under the influence of a weak potential.For larger Ti, oscillations that are neither purely librational nor purely rotationalappear on the correlation functions. The spectra for reasonably short zi recall thoseobserved in liquids made of sterically symmetrical molecules in a weakly interactingsolvent .zR. G. Gordon, J. Chem. Phys., 1965,43,1307.Y . LeRoy, E. Constant and P. Desplanques, J . Chim. Phys., 1967, 64,1499.R. G. Gordon, J. Chem. Phys., 1966,44,1803.H. Bateman, Tables of Integral Transforms, (McGraw Hill, 1954).Y . Rocard, J. Phys. R d . , 1933, 7, 247 (in the numerator of his expression (8), the plus signshould be replaced by a minus sign).H. Shimizu, J. Chem. Phys., 1965,43,2453 (see formula (5.16) with (5.13)).C . Brot, Compt. rend., 1968,266,72.K. F. Herzfeld, J . Amer. SOC., 1964, 86,3468 (with the same correction as in Rocard).' C. Brot, J. dephys., 1967,28,789.lo C. Brot, Chem. Phys. Letters, 1969, 3, 319.l 1 E. Bauer, Cahiers de Phys., 1944, no. 20, 1.l 2 H. Frohlich, Theory of Dielectrics, (Clarendon Press, Oxford, 1949).l3 B. Lassier, C. Brot, G. W. Chantry and H. A. Gebbie, Chem. Phys. Letters, 1969, 3,96.l4 B. Lassier and C. Brot, Chem. Phys. Letters, 1968, 1,581.l5 G. W. Chantry and H. A. Gebbie, Nature, 1965,208,378.l6 Y. LeRoy and E. Constant, Compt. rend., 1966,262,1391.l7 N. E. Hill, Proc. Phys. SOC., 1963,82,723

ISSN:0366-9033    

DOI:10.1039/DF9694800039

出版商:RSC    

年代:1969

数据来源: RSC

摘要:

Lattice Dynamics and Restricted Rotation in Molecular CrystalsBY P. H. MARTIN AND S. H. WALMSLEYWilliam Ramsay and Ralph Forster Laboratories, University College, LondonReceived 5th June, 1969The problem of coupled rotors in molecular crystals is reviewed in the limits of low and highbarrier to rotation. An approximate one-molecule potential is suggested. The phase propertiesof mixed crystals of para- and ortho-hydrogen are discussed using this model.The conventional theory of lattice dynamics deals with the problem of thevibrational modes of crystals. In its simplest form, the harmonic approximationis made and the equations of motion are uncoupled by transforming to the appro-priate set of normal displacement coordinates. The effects of anharmonicity, ifthey are sufficiently small, may be taken into account by using perturbation theory.Such a scheme involves the continued use of displacement coordinates and impliesthat the nuclei are each strongly localized in the region of the equilibrium configura-tion.For molecular crystals in particular, the molecular units are held togetherby much stronger forces than the intermolecular interactions responsible for thecondensed state. It is possible to envisage a situation in which the molecular orienta-tion is not strongly fixed in the crystal, so that the molecules either can execute nearlyfree rotations or have an intermediate behaviour in which they are in some sensedisordered over a number of equivalent configurations. The methods of latticedynamics are no longer appropriate and it is the object of this paper to discusspossible procedures.COUPLED ROTORS IN LOW AND HIGH BARRIER LIMITSTo simplify the problem, it is assumed that the molecules are rigid and that thecentres of mass are stationary.The Hamiltonian H then has the general form givenin eqn. (1) :H = CLi(mi) + V(. . .,mi,. . .) (1)iLi(col) is the kinetic energy operator for rotation of molecule i, with cof standing forthe set of Euler angles connecting the principal axes of inertia of the molecule with aconvenient crystal fixed Cartesian axis set. V is the potential energy operator whichis here assumed to be a sum of pair potentials :v = SC C fij(mi,mj)* (2)i j#iThe eigenfunctions andIeigenvalues of H are not readily found for the generalcase, but the problem may be simplified in a number of limiting situations. Thecritical property of the potential energy V is the difference between its maximum andminimum values, which gives a measure of the barrier to rotation. If the barrier islow with respect to rotational excitation energies, the potential may be treated as aperturbation.The unperturbed system is equivalent to a lattice of non-interacting450 RESTRICTED ROTATIONrotors, and the wave equation is separable in the angular coordinates mi. Taking alinear molecule as an example, the molecular wavefunctions are the sphericalharmonics :Y J M ( e l , 4 i ) = YJM(i), J = 0,1,2,. . ., (3)M = -J, . . ., 0,1,. . .,+J.and the corresponding energies,where I is the molecular moment of inertia.Unperturbed crystal wave functionsare given by products of functions of type (3), one term for each molecule. Thecorresponding energy is the sum of the individual molecular energies.The evaluation of the matrix elements of the perturbation operator is considerablysimplified if the pair potentials are themselves expanded in spherical harmonics asin eqn. (5) :E,(i) = (h2/8n21)J(J+ I), (4)The resulting treatment has exactly the same form as the theory of excitons inthe tight binding limit for the electronic excited states of molecular crystals. Thisapproach has been developed by van Kranendonk and applied to the problem ofsolid hydrogen.In most molecules, the rotational excitation energies are of order 1 cm-l and thelow barrier limit is only of effective use for molecules in which only hydrogen atomsare rotating. The problem is also complicated by the effects of nuclear spin.Whena molecule contains more than one nucleus of the same type, there exist nuclear spinisomers which are not easily interconverted. In this way, solid hydrogen as normallyprepared is a mixed crystal of 1 part para- (in which all molecules have an even valueof the rotational quantum number J ) and 3 parts ortho- (odd J ) hydrogen. For thisexample, it is possible to separate the two isomers and a continuous range of solidsfrom pure para- to pure ortho-hydrogen may be prepared. A theoretical problemwhich remains is that the exciton problem has not been solved for the case in whichthe molecular ground state is degenerate, as for pure ortho-hydrogen.Methane hasthree nuclear spin isomers and no method of separating them has yet been found.A crystal of methane is therefore always the appropriate equilibrium mixture.The problem is also soluble in the limit in which the barrier opposing rotationbecomes high. If there is a single distinct deep minimum, the system becomes equiva-lent to the convential model of lattice dynamics discussed in the first section. TheEuler angles are replaced by suitable displacement coordinates and the moleculesexecute torsional oscillations about the minimum. It is no longer necessary torestrict attention to orientation coordinates, and the effect of centre of mass displace-ments and internal molecular vibrations is readily added.EFFECTIVE ONE MOLECULE POTENTIALTo make progress in the region between the two limits, a different approach isrequired.It is here suggested that this may be achieved through an approximationto the potential energy which uncouples, or at least considerably simplifies, theproblem. One possibility is an effective one-molecule potential. This is clearlymore closely related to the rotational exciton than to the lattice vibration limit.However, calculations and experimental results suggest that in some cases torsionallattice vibrations are not strongly dispersed with respect to the wave vector of thecoordinate, and a localized vibration may be usefulP. H. MARTIN AND S. H. WALMSLEYIn this method the general Hamiltonian is replaced by one of the form :H = C(L,+ 6) = C H , ,51i iand it is required to solve for the eigenvalues and eigenfunctions of Hi.Vi involvesthree coordinates (or two for a linear molecule) and the problem is not, in general,soluble in closed form. A good approximation to the eigenfunctions may be foundby using a suitable set of basis functions as a linear variation set: rotational orvibrational wave functions or solutions of Mathieu's equation may be useful indifferent circumstances. The form of V, is dependent on the eigenfunctions of Hiand a cyclic procedure is used to bring the two parts to self-consistency.One step in the cycle has the following stages. As an example, it is assumed thatthe crystal contains linear molecules and that the free rotor wave functions are beingused as a basis set.The one molecule potential Vris derived from the previous stepin the cycle. The corresponding one molecule Hamiltonian is If: defined in (6).The approximate eigenfunctions of this operator have the form :$xi> = C C J M p Y J M ( i ) (7JMwith corresponding eigenvalues E;. The ground-state wave function will be denotedby +t(i). The next approximation to the potential is found by averaging the truepotential energy over the ground-state wave function of all molecules except one :The cycle is then repeated to self-consistency.The averaging procedure described in (8) may be elaborated to take account oftemperature and concentration of nuclear spin species. For the first of these, theaverage on the right-hand side of (8) is repeated for excited states t,hi(j) and the poten-tial is a linear combination of the averages, in which the coefficients reproduce theBoltzmann distribution for the particular temperature.For the second, the wholeprocess is repeated for each nuclear spin species and an overall linear combination,determined by the concentrations, made to form the final potential.PHASE TRANSITION I N MIXED CRYSTALS OF PARA- A N D ORTHO-HYDROGENAs an example, the theory is applied to solid hydrogen. The concentration ofortho- and para-molecules is allowed to vary and the two isomers are treated asseparate molecular species. The range considered is that below 10 K, and it isassumed that attention may be confined to J = 0 wave functions for para-hydrogenand J = 1 wave functions for ortho-hydrogen. Two crystal structures are considered,in which the molecules are hexagonal or cubic close-packed.For the hexagonalstructure the axis of quantization of the J = 1 functions is chosen to be parallel tothe crystal six-fold symmetry axis; and for the cubic structure one of the four-foldaxes is chosen. Within these simplifying approximations, the rotational wave func-tions remain eigenfunctions of the successive Hamiftonians. In this way, the co-efficients of the wave functions in the first and higher approximations take accountonly of thermal populations and of ortho- and para- concentrations.The effective one molecule potential at the nth iteration has the form :Vn(i) = (A+BPx)Y,,,(i)+(C+ DPx)Y,,,(i), is>in which P is the concentration of ortho-hydrogen ; A , B, C, D are lattice sums, andfor the cubic structure B=C=O.x is a measure of the splitting of the A4 = 52 RESTRICTED ROTATIONand M = & 1 levels belonging to J = 1, arising from the previous step in the iteration.Explicitly,in which E = E,-E,, where the subscript refers to the value of M. The conditionthat self-consistency has been reached may be shown to have the form,where s depends on the lattice sum C, and t on the product DP.x = [ 1 - exp (E/kT)]/[ 1 + 2 exp (E/RT)], (10)x = [l --s exp (tx/T)]/[l+2s exp (tx/T)), (11)Y1FIG. l.--Solutionsofeqn.(ll): - - - y ; = x;--y = [l--sexp(tx/T)]/[1+2sexp(tx/T)] with(1) PITsmall, (2) PIT large.For the hexagonal case, t is positive and s is approximately unity, so that the onlyreal root x of (11) is close to zero.In the cubic structure, t is negative and s isidentically unity. As shown in the figure, for sufficiently high values of PfT, eqn.(1 1) may have two solutions in addition to x = 0.has been used,will be published elsewhere. Here a brief statement of the qualitative nature ofthe results is given. At the top end of the temperature interval, x = 0 is the onlyroot for both structures and there is no splitting of the J = 1 states. Which of thestructures is more stable is determined by the difference of their total free energies.Here it can only be noted that, for equal nearest neighbour distances in both structures,the static lattice energy together with rotational free energy slightly favour thehexagonal structure. As the temperature is lowered, for a sufficiently high ortho-concentration, there are three cubic possibilities, and the lowest free energy is asso-ciated with the cubic structure in which the M = 0 level is lower in energy than theM = fil.This calculation does therefore reproduce qualitatively a salient feature of solidhydrogen, i.e., for crystals containing greater than 58 % ortho-hydrogen a transitionfrom hexagonal to face-centered cubic structure is ~bserved,~ at temperatures between1 and 3 K, the transition temperature being a function of the ortho-hydrogen con-centration.Full details of this work, in which the potential of NakamurP.H . MARTIN A N D S. H . WALMSLEY 53The structure which is favoured by our model is strictly tetragonal since the mole-cules are, in effect, partially ordered about a single four-fold direction. This isprobably a consequence of the model, in that there is a restriction to an effective onemolecule field. There is some evidence that the cubic structure is a more complicatedone with four molecules per unit cell, analogous to that of carbon dioxide.6 Workis in progress to make more quantitative estimates of the phase transition data andto examine possible extentions of the one molecule effective field.One of us (P. H. M.) acknowledges the award of an S.R.C. Studentship, duringthe tenure of which this work was carried out.l M. Born and K. Huang, DynamicaZ Theory of Crystal Lattices (Oxford University Press, 1954).J. van Kranendonk, Physica, 1959,25,1080.D. A. Oliver and S. H. Walmsley, Mof. Phys., 1968, 15,141.T. Nakamura, Progr. Theor. Phys., 1955,14,135.A. F. Schuch, R. L. Mills and D. A. Depatie, Phys. Rm., 1968,165,1032.J. C. Raich and H. M. James, Phys. Rm. Letters, 1966,16,173

ISSN:0366-9033    

DOI:10.1039/DF9694800049

出版商:RSC    

年代:1969

数据来源: RSC

摘要:

Rotational Excitations in Solid Hydrogen and Deuteriumin the Ordered StateBY ISAAC F. SILVERA, WALTER N. HARDY AND JOHN P. MCTAGUEScience Center, North American Rockwell Corporation,Thousand Oaks, CaliforniaReceived 16th June, 1969New rotational Raman spectra have been obtained in solid ortho-hydrogen and para-deuterium.In the orientationally ordered state a total of five librational lines was observed in the region 5 +30cm-' which is consistent with a previous proposal that the generally accepted Pa3 structure be replacedby a rhombohedra1 distortion of Pa3, namely, 2 3 . However, the J = 0 +2 spectra of para-hydrogenand ortho-deuterium impurities have four lines, three of which seem to fit the Pa3 structure. Weconclude that a consistent picture of the ordered state has to include additional interactions such asa libron-phonon coupling.Crystalline hydrogen and deuterium, as distinct from most solids, exhibit well-resolved rotational spectra.This is a consequence of the fact that the free moleculerotational energy spacings are much larger than the anisotropic forces encountered inthe condensed phases. At the low temperatures of the solid (0+-15 K) only thelowest rotational levels of the respective species are populated : J = 1 for ortho-hydrogen (0-H,) and para-deuterium (p-D,), and J = 0 for para-hydrogen (p-H,) andortho-deuterium (0-D,). The rotational and vibrational spectra of the even Jspecies, p-H2 and 0-D,, have been studied extensively by the Toronto group,l yieldinga detailed description of the interactions in the solid when only low concentrations ofthe J = 1 species are present.We report here our studies of the rotational Ramanspectra in high purity ortho-hydrogen and para-deuterium. There are two basictypes of transitions that one observes : the AmJ transitions within the J = 1 manifoldand the J = 1-+3 transitions. In practice, there is also a small quantity of the even Jspecies present as an impurity and the resulting J = 0-2 spectra can serve as anadditional source of information.The ground state of the solid is complicated by the multiplicity of the J = 1molecular level and the non-zero expectation value of the electric quadrupole momentwhich leads, at sufficiently low temperatures (T, = 2.8 K for 0-H, and 3.8 K forp-D2), to an ordering of the molecular axes analogous to spin ordering in an antiferro-magnet.Above the critical temperature, the molecular centres form an h.c.p. lattice and themolecular axes have no preferred orientation.Equivalently, the three m levels(0,k 1) are degenerate. Below T, the anisotropic intermolecular interactions causethe molecules to assume preferred orientations with respect to the crystalline axes andthe molecular field gradients partially lift the degeneracy of the mJ levels. Accom-panying the orientational transition is a change of crystal structure to one in which themolecular centres are located on an f.c.c. lattice., Several theoretical calculationshave predicted a Pa3 space group, a result consistent with later neutron diffractiondata.4 In this structure there are four molecules per unit cell; the axes of the51. F.SILVERA, W. N. HARDY AND J . P. MCTAGUE 55dumb-bell shaped mJ = 0 ground state wave functions are aligned along the fourf.c.c. body diagonals. The crystal is divided into four equivalent sublattices with thedirections of quantization of J on a given sublattice being parallel. In the disorderedstate the rotational excitations are essentially localized on a single molecule, the life-time, or spectral width, being determined by the orientational fluctuations of theneighbouring molecules. In the ordered state, the periodicity of the molecularorientations results in rotational excitations of a non-localized collective nature.Those excitations in which only mJ changes are called angular momentum waves, orlibrons, and are closely analogous to spin waves in ordered magnetic insulators.Theexcitations involving a change in J are called rotational waves and have been studiedextensively in para-H2 by Van Kranendonk and co-w0rkers.lWe note that if the intermolecular forces were strong enough to appreciably mixthe J states then the division into librational and rotational waves would no longer beappropriate. In the limit of very strong crystal fields, it becomes meaningful to usethe classical coordinates of the molecular axes, and the resulting excitations are thenthe classical librational waves studied by Walmsley and Pople and others in C 0 2 andN2. In the classical limit, the number of librational modes (for the Pa3 space group)is eight, the same as for the AJ = 0 quantum mechanical modes in o-H, and p-D,.This results in a strong formal resemblance between the Q-M and classical librationalmodes : their symmetries and degeneracies have a one-to-one correspondence.Further enhancing this resemblance is the fact that at k = 0 the ratios of the classicallycalculated frequencies are equal to those of the non-linear spin wave treatments,EXPERIMENTALThe samples of o-H2 and p-D2 were prepared from normal gas using an adsorptioncolumn of the type described by Depatie and Mills.6 The initial purities as measured by aTABLE 1 .-EXPERIMENTALLY OBSERVED FREQUENCIES IN SOLID H2 AND D2para-deuterium (97 %) ortho-hydrogen (F97 a%)temp.[Kl freq.shift [cm-11 assignment and comments temp. [Kl freq. shift [crn-l] assignment and comments4.21.5-37.1 f l173.8 33'300.7 f3*8.8 f l11.2f115.1 f l22.5 f 129.9 f 2172.3 f 2 t174.7 f 2181.3 f 2186.6 f 2296.2 &2t303.6 f 2308.7 f 2Orientational band indisordered stateOptical phonon (E2Jimpurity transitionsJ = 0+2 : o-DZJ = 1-3 rotationalbandE'(2) LibrationalEg(2) modes.A@ TentativeE,(2) assignmentAQ for R3.J = 0+2 localizedimpurity transitionsJ = 1 + 3 non-localizedrotational band4.21.51.341.3438.2 & 1348.5 5 3586.3 f 36 . 2 f l8.0 f 111.3i-116.8 f 121.0 f 2347.5 f2t350.2 f 2354.6 f 2359.1 f2582.8 f2t588.6 f 2593.0 f2Orientational band indisordered stateOptical phonon (E2Jimpurity transitionsJ = 1+3 rotationalbandJ = 0-2 : P - H ~E'(2) LibrationalEJ2) modes.A, TentativeEJ2) assignmentA@ for R3.J = 0+2 localizedimpurity transitionsJ = 1-3 non-localizedrotational band* peak values of asymmetric lines. t error in relative frequencies within the band = rt0.5 cm-'56 ROTATIONAL EXCITATIONS I N SOLID HYDROGENthermal conductivity bridge were around 98 %. The solid samples used in the experimentshad to be free of cracks that could scatter laser light directly into the monochromator,otherwise the low-frequency Iibrational lines were obscured by spurious lines (grating ghosts)and by spillover of laser light due to the instrumental slit function.For hydrogen, theortho-para conversion rate in the solid is about 2 %/h which allowed only a short time forthe experiments to be performed; in deuterium, the rate is approximately twenty timesslower.The growth in a reasonable length of time of samples clear enough for resolutionof the low-frequency Raman features presented the greatest experimental difficulty and forthis reason is described in some detail.The samples were first rapidly solidified by lowering the Raman cell into the liquid helium,at which point the valve to the sample bulb was closed (fig. 1). The cell was then liftedabove the liquid helium, the sample remelted, and the cell then lowered very slowly untilsolidification initiated at the bottom of the tube. The growth was allowed to proceed at a1 , To precision pressure gouqeFIG.1.-Experimental apparatus for right angle Raman scattering and crystal growth. The laser beam,after passing through the sample is reflected up and out of the dewar by the second surface mirror.No optical windows are required in the dewar. Details of the crystal growth region are shown inthe enlarged diagram. The copper heat conductor is designed to cool the sample from a small spotat the bottom so that growth is properly initiated.rate of 1 to 4 cm/h (a typical sample length was 2 cm). At the completion of solidification,the upper surface of the sample was concave. If at this stage the cell was further cooled,cracking in all parts of the sample generally resulted. The cracking was particularly severein p-D2. To avoid this problem, which we suppose was due to sticking of the sample to theglass walls of the cell, the sample was held at a temperature just below the triple point for10-30min, the temperature being monitored via the vapour pressure. The shape of thesample surface eventually became convex and the top 0.5 cm of the sample pulled awayfrom the walls.Careful cooling then resulted in a sample that was completely crack freein the upper region. Examination of the crystals in the hexagonal state with crossedpolarizers suggested that the samples consisted of single crystal domains with dimensionsof a few millimeters and having the c-axis preferentially in the vertical direction. Clearsamples of para-hydrogen were very easy to grow with this method, and the fist such attemptresulted in what was very likely a large single crystaI.To ensure good thermal contact, afew cm of liquid He was condensed in the Raman tube, the vapour pressure of which servedas the thermometer.The exciting source was the 5145A line of a Coherent Radiation Laboratories Model52 argon ion laser operating in the power range 0.2-+0.7 W. Right-angle scattering waI . F . SILVERA, W . N . HARDY AND J . P . MCTAGUE 57observed using a Spex model 1400 double monochromator and a cooled I.T.T. FW-130photomultiplier whose output was detected with a picoammeter.The ordering transitions in H2 and D2 show irreversible effects and for this reason thetemperature was always changed monotonically. Spectra were normally taken only fordecreasing temperature and samples brought back through the transition were annealednear the triple point before taking additional spectra.RESULTS AND DISCUSSIONTheoretical calculations of the frequencies and degeneracies of the librons for thePa3 structure '-l O predict three Raman active modes (E, + 2T,) in the frequency range-5-20 cm-l.However, earlier observations l1 taken with lower signal to noiseshowed four distinct lines with one of them being relatively broad and asymmetric.In order to explain the number of lines, it was suggested that the structure was oflower symmetry. The simplest distortion of the Pa3 structure maintaining the inver-sion site symmetry gives the rhombohedra1 space group R3 (Czl). In this structurethere are still four molecules per unit cell ; however, the molecule on the trigonal axisFra.2.-Libron spectrum in para-deuterium ofpurity 97.1 %. The insert is at higher gain showingthe splitting of the broad line at -21 cm-l. (b) Thesame spectral region in the disordered hexagonalstate. The sharp feature at -37 cm-1 is due to aRaman-active phonon. In (b) the gain is 4.8times that in (a).is no longer equivalent to the other three. There are two distinct ways in which thePa3 structure could distort to the R3 space group. The first is to have a compressionor elongation of the f.c.c. positions along one of the <1 11) axes ; however, this doesnot strongly affect the energy levels. The second is to rotate equivalently the quantiza-tion directions of three of the sublattices which can more readily change the energ58 ROTATIONAL EXCITATIONS I N SOLID HYDROGENlevels.This structure would result in a partial lifting of the degeneracy of the three-fold Tg modes of the Pa3 space group into doublet E, and singlet A , modes. Fivedistinct Raman active lines are then predicted. Fig. 2a shows the more recentlyobtained data for p-D2 in which five lines are now clearly resolved. Additionalspectra were recorded as the temperature was lowered from (T,-O.l K) to N 1.4 K.Although there was a progressive line narrowing, no shift in the line frequencies wasobserved. Analogous results have been obtained for o-H, and both sets of spectralend support to the earlier proposed Rs space group? In order to compare totheory, a small perturbation was assumed so that the centres of gravity of the observedlines could be correlated with the theoretical line positions for a Pa3 space group,An earlier such fit,ll for which the highest frequency line was not observed due to lowsignal-to-noise, gave excellent agreement.For the new data, the centre of gravity ofthe highest two lines apparently lies a few cm-l higher than required for good agree-ment. The experimentally observed line frequencies for both H2 and D2 are given intable 1. Although the proposal of the R3 space group explains the number of ob-served lines, there are other possibilities within the Pa3 structure which must still beconsidered. Since the samples always contained at least 2 % J = 0 species, it waspossible that some of the lines were due to the presence of the impurities.However,over the concentration range 97-93 %, no significant change was observed in the relat-ive intensities of the lines. Another possibility is a dynamical libron-phonon inter-action. Although rough estimates lo suggest that the phonons would have a smalleffect on the libron dispersion relations, this possibility cannot be discounted.In the orientationally disordered hexagonal state the low-frequency spectraexhibit no sharp features (fig. 2-b) but rather a broad, approximately Gaussian shapedline whose width is related to the time dependence of the molecular reorientation.(The sharp line at approximately 37 cm-l which disappears below T' is attributed to theRaman active phonon of the hexagonal state and will be discussed at a future time.)A preliminary analysis indicates that for p-D2 0.4 K above T, the Stokes part ofthe experimental spectrum can be fitted to a Gaussian function having a width para-meter CT N 10 cm-l and shifted by -2 cm-l from the origin. Moriya and Matsuda l2and also Harris and Hunt l3 have calculated the second moments of the single mole-cule reorientation frequency distribution in the high-temperature limit.However,frequency distribution required for the Raman spectrum is that of the sum of thepolarizabilities of the molecules, and although related is not the same. Calculationsare now in progress. N.m.r. relaxation data l4 show that in the vicinity of v 21 0, thespectral density increases as the temperature is lowered which seems to be inconsistentwith the picture of a Gaussian line moving away from the origin as the temperature islowered.Further experiments are in progress to determine the temperature depend-ence of the shift and of the width.The transitions involving a change in J behave in a qualitatively similar manner,In the high temperature phase (fig. 3-b), the J = 1+3 line is broad and asymmetric.In the ordered state, the ground level sharpens up into the I J = 1 , mJ = 0 ) state.The excited states are again collective with a multiplicity 28 since there are four sub-lattices [4(W+ l ) ] . Preliminary calculations of the roton levels for the Pa3 structurecarried out elsewhere l5 predict 10 distinct lines. The observed spectrum (fig. 3-4shows but three distinct lines.Further interpretation will require the results ofintensity calculations now in progress.Finally, we discuss the impurity J = 0+2 transitions shown in fig. 4. Theintegrated intensity of this transition relative to the J = 1 4 3 band serves as a con-venient monitor of the sample purity. The concentrations determined by the intensityratio for the disordered state I(J = 0+2)/Z(J = 1 +3) = 5/3(c/l -c), where c is th1. F. SILVERA, W. N. HARDY AND J . P. MCTAGUE 59concentration of the J = 0 species have been found to be in good agreement withconcentrations implied by the transition temperature,16 T,, and by measurements ofthe thermal conductivity of the gas./ \ d L L-I--I--l ._ - 1 - u _I270 2 8 0 290 300 310 320 ' 3 3 0 340C l l i 'FIG.3.-J= 1+3 transitions for para-deuterium (97 % purity) : (a) below theordering temperature T,; (b) above Tc.The intensity scales are approximatelyequal in (a) and (b).-.a I t t1 6 k - L _ i . _ l - . 1 I - - L-l.--i170 180 190 700T.4.2 KL--' - - I - r d _ - - I160 180 2 0 0 2 2 0c m - 'FIG. 4.-J = 0+2 impurity transitions forpara-deuterium (purity 97 %) : (a) below Tc.The arrows indicate the expected positions ofthe lines for a C3j point symmetry; (b) aboveTc. In (a) the gain is approximately fourtimes greater than in (b).The modes associated with the isolated impurities are localized rather thancollective, and their spectra thus afford a probe of the local site symmetries andinteractions. In the Pa3 structure, the site symmetry is C3* for each of the fourequivalent sublattices.We have calculated the J = 0-2 impurity spectrum assumingnearest neighbour quadrupole-quadrupole interactions and using the J = 0-2central frequency determined from Raman spectra of o-D,. This calculation predictsthree lines whose positions are indicated with arrows in fig. 4-a, in reasonable agree-ment with three of the four observed lines. Similar agreement with the same threelines was found with hydrogen. For the R3 space group, the inequivalent site hasC,, point symmetry, and should exhibit the above calculated spectrum. The threeequivalent sites have Ct point symmetry and the J = 2 level should be split into 5non-degenerate states. This is difficult to reconcile with the observed spectrum.Furthermore, from intensity measurements above and below T,, essentially all of thesites must contribute to the 4 observed lines.The absence of additional resolve60 ROTATIONAL EXCITATIONS I N SOLID HYDROGENlines other than the appearance of the unassigned line at 181 cm-I seems to beinconsistent with the large splitting of the libron modes.Although, taken alone, the new data on the librational waves appear to corroboratethe previously proposed R5 structure, consideration of the J = 0-+2 impurity spectracasts some doubt on this interpretation. If indeed the Pa3 structure is correct, thenthe theories describing the librational waves are incomplete, the most probableweakness being the assumption of a rigid lattice which excludes rotation-translationcoupling.To proceed further in the interpretation of the Raman scattering data, itwould be helpful to have a calculation of the libron-phonon coupling and/or aprecision determination of the crystal structure.We thank Lloyd Ahlberg and John Curnow for skilled technical assistance.A review and bibliography can be found in J. Van Kranendonk and G. Karl, Rev. Mud. Phys.,1968, 40,531.R. L. Mills and A. F. Schuch, Phys. Reu. Letters, 1965,15,722; A. F. Schuch and R. L. Mills,Phys. Rev. Letters, 1966, 16,616.J. Felsteiner, Phys. Reu. Letters, 1965, 15, 1025; H. M. James and J. C . Raich, Phys. Reu.,1967, 162,649.K. F. Mucker, S. Talhouk, P. M. Harris and D. White, Phys. Rev. Letters, 1966, 16,799.S. H. Walmsley and J. A. Pople, Mul. Phys., 1964,8,345.D. A. Depatie and R. L. Mills, Rev. Sci. Instr., 1968, 39, 105.S. Homma, K. Okada and H. Matsuda, Progr. Theor. Phys., 1966,36,1310; 1967,3$, 769.H. Ueyama and T. Matsubara, Progr. Theur. Phys., 1967, 38,784.J. C. Raich and R. D. Etters, Phys. Rev., 1968, 168,425.lo F. G. Mertens, W. Biem and H. Hahn, 2. Phys. 1968, 213,33; 1969,220,l.l 1 W. N. Hardy, I. F. Silvera and J. P. McTague, Phys. Rev. Letters, 1969, 22,297.l2 T. Moriya and K. Motizuki, Progr. Theur. Phys., 1957,18,183.l3 A. B. Harris and E. Hunt, Phys. Rev. Lefters, 1966, 16,845.l4 L. I. Amstutz, H. Meyer and S . M. Myers, to be published.l5 J. Raich, private communication.l6 A. F. Schuch, R. L. Mills and D. A. Depatie, Phys. Rev., 1968, 165,1032.l7 W. Biem and F. G. Mertens, Bull. Amer. Phys. Suc., 1969, 14,335

ISSN:0366-9033    

DOI:10.1039/DF9694800054

出版商:RSC    

年代:1969

数据来源: RSC

摘要:

DISCUSSION REMARKSDr. R. J. Elliott (Oxford University) said: It is interesting that Trevino is able topresent his w(q) curve in a Brillouin zone, where qmax = 2n/a and a is the C-C spac-ing along the axis of the helix, rather than an 2n/na, where na (with n = 15 or 13)is the repeating distance. At first sight one might expect gaps in the m(q) curve atthese boundaries. The fact that there are none may arise in two ways. (a) Thesystem has a symmetry greater than that for a repeat distance of nu, or, (b) someforce constants have been neglected so that to this approximation the symmetryappears higher than it really is.One can define a new symmetry groupisomorphous with the translation group where each element consists of a translationalong the c-axis followed by a rotation of the helix through an angle 271 x (7/15) (or271 x (5/13) in the other phase).Excitations will still be described by a q value-nowin going from C to C the displacements in a mode are related by this operation, i.e.,Teflon appears to be the first case.u(l+ 1) = Ru(1) exp (iqa),where R is the rotation just described as well as the usual wave-like change exp (iqa).A similar theory has been proposed for excitations (electrons and spin waves) in rareearth metals which show spiral magnetic order. It may be noted that this symmetrygroup applies even when the helix is incommensurate (n is an irrational number).A second point of interest arises from the softness of the modes near the zoneboundary. In fact, the ratio co(qmax)/cl/'max is extremely small.Such low frequencymodes indicate either (a) a tendency towards instability as in Cochran's theory offerroelectrics or (b) some further hidden symmetry in the problem. It is possiblethat both types of mode will appear in this material and I would ask the nature ofmodes near qmax.(a) The phase change from (7/15) to (5/13) corresponds to an unwinding of thespiral. Thus, at T, the mode, where m2+0 because of anharmonic effects, shouldhave neighbouring C motions in opposite directions parallel to z and around thecircle in opposite directions in the plane perpendicular to z. These would in factappear to be just at q = n/a. It would be interesting to study the transition. (b)Zero frequencies can appear for all T if the modes correspond to a motion whichclearly has no restoring force.Such extra symmetries indicate that the ground of thesystem is degenerate-this idea has been used in Goldstone's theorem in relativisticquantum mechanics and in its extension to systems in solid state physics. In latticedynamics the best known trivial result is that a+O as 4-0 for acoustic modes sincethey correspond to uniform translation of the whole lattice. In this case there isanother possible mode of interest, uniform rotation of a helix. This will involve noenergy if interactions between helices are neglected. Such uniform rotation is atransverse mode with q = 14n/15a also very near qmax, and its frequency will onlyreff ect forces between helices.Dr. S . F. Trevino (Army Materials and Mechanics Research Center, Mass.): Withrespect to Elliott's first observation, the interpretation is correct.Secondly, thepolarization vector at q = n/a of the mode which was measured fit his description ofthe type of mode which could lead to an unwinding of the helix. It is our intentionto investigate the behaviour of this vibration through the transition. Finally, azero frequency will occur corresponding to uniform rotation if interchain forces are662 GENERAL DISCUSSIONneglected. This frequency however occurs for q = 0, not 14n/15a. This can beseen from inspection of the equation he gives relating the displacement of adjacentparticles. The motion would require G(l) to be tangent to the circle described by theparticle position, and that u(Z+ 1) be rotated by R and move in the same direction asu(l), i.e., q = 0.This motion corresponds to the lowest acoustic mode which is nowshown in the figure.Dr. D. W. McCall (Bell Telephone Lab., N.J.) said: In connection with Elliott’scomment concerning the indications of instabilities by the existence of soft modes, itis significant that polytetrafluoroethylene exhibits a phase transition near 40°C.N.m.r. studies of this transition indicate the onset of translational motions parallelto the chain axes.Dr. G. Zerbi (Politecnico-Milano) said: Several minima have been found in thecalculation of the conformational potential energy of Teflon. Minima with increasingstability have been calculated for torsional angles of -60, -90, and - 165°C.Thedeepest minimum at - 165°C (which corresponds to the helical conformation experi-mentally observed) is separated by a very low barrier from the one at -195’ whichcorresponds to the enantiomorphous structure. It is then likely that above 20°Cthe chain consists of short segments of right-handed and left-handed helices joinedby bonds in trans conformation. My question refers to Teflon but may be applied toother helical polymers : how long should the helical segments be in order to observethe associated phonons ?Prof, J. A. Jan& (Krakow) said: Why did not Hyde et al. pass through the phasetransition point below the plastic phase? It would be interesting to see not onlythe jumps at the melting point, but also what happens when the crystal loses itsplastic properties.Dr.A. J. Hyde and Dr. J. N. Sherwood (University of Strathclyde) said : RegardingJanik’s question about the non-plastic to plastic transition, we are examining thistransition as the next stage of the work. The observations are more difficult than onthe melting transition since one has to produce a crystal of the low temperature phaseand observe what happens as the temperature rises. All attempts to pass through thetransition by cooling have given rise to shattered crystals (presumably of the low-temperature form) that are useless for scattering experiments.Dr. M. 0. Norris (University of Kent at Canterbury) said: As a co-author of oneof the papers (ref. (13)) mentioned by Hyde, I would comment on anomalies in resultsobtained with succinonitrile.We found two anomalies, only one of which may ormay not be associated specifically with the melting point. The first is concerned withthe differences in reorientational correlation time deduced from our values of protonspin-lattice relaxation time and from two sets of dielectric relaxation results. Wehave suggested that these differences may be due to the three distinct types of motionpossible in this flexible molecule contributing differently to the nuclear magnetic andthe dielectric relaxation processes. These molecular motions apparently occur overthe whole of the temperature range from the transition point to the melting pointand do not, therefore, represent any effect specifically connected with the meltingprocess.Another anomaly we encountered seems to bear some relation to the diffusion ofdisordered regions within the lattice, suggested by Hyde as an explanation of thGENERAL DISCUSSION 63high values of Landau-Placzek ratio he obtained.This anomaly may or may not beassociated purely with the melting process. Our values of proton-spin-lattice relaxationtime Tl close to the melting point turn over in an anomalous way and show somefrequency dependence. Values of spin-lattice relaxation time in the rotating frameT , , are less than Tl over most of the phase. When values of TIP are compared withTl in the anomalous region they appear to be in the ratio of resonance frequencyto the power 4, as compared with the more usual power of 2. We have thereforeattributed this to a " defect diffusion " process rather than to the more usual mole-cular diffusion process.The data are not as clear-cut as we would like and it would be useful to havevalues of Tl at lower resonance frequencies.This, together with values of Tlp inhigher values of rotating magnetic field, should show clearly if this defect diffusionmodel explains the anomaly or if it is associated solely with the onset of melting.Dr. A. J. Hyde and Dr. J. N. Sherwood (University of Strathclyde) (communicated) :During the course of the meeting we were informed by Dr. Hubert Fontaine of LilleUniversity that he had recently made extensive measurements of the velocities oflongitudinal and transverse waves in single crystals of succinonitrile by conventionalultrasonic methods right through the plastic range and had obtained values of soundvelocities about 5 % lower than our values.In connection with the dip in scattered intensity just before the melting point, arecent paper by Della Monica on the electrical conductance of very low concentra-tions of electrolyte in the plastic phase of sulpholane I'"-"..:] shows ananomaly just before the melting point which appears to parallel closely the change indielectric constant of this materiaL4 The available measurements of the dielectricconstant of succinonitrile do not show such a dip, but one set of measurementsdoes not go higher than 50°C and the other does not show experimental points.6 Ifthere is a dip in the dielectric constant of succinonitrile as there is in sulpholane thenthis would parallel the scattering change./ \CHZ-CHZProf.J. A. Janik (Krakow) said: Is it possible to think about a different theoreticalapproach for solid a-N,-an analogous one to that applied for solid hydrogen andfor the low-temperature phase of solid methane. I am thinking of an approach inwhich molecular interaction is taken as multipole-multipole interaction (quadrupole-quadrupole, or octupole-octupole).Prof. 0. Schnepp (University of S. California, L.A.) said: In reply to Janik'squestion concerning solid a-N,, we were not successful in accounting for the librationalfrequencies in terms of the quadrupole-quadrupole potential term. A similar con-clusion was arrived at by Walmsley and Pople in their work on solid CO,.B.I. Hunt and J. G. Powles, Proc. Phys. Soc., 1966, 88, 513.H. Fontaine and C. Moriamez, J. Chim. Phys., 1968, 65,969.M. Della Monica, J. Amer. Chem. Soc., 1969, 91, 508.U. Lamanna, 0. Sciacovelli and L. Jannelli, Gazz. Chim. Ital., 1966, 96, 114.ref. (6), our paper.ref. (7), our paper64 GENERAL DISCUSSIONDr. C. Brot (Lab. de Chim. Phys., Orsay, France) said: The advocates of the atom-atom potential method (e.g., Kitaigorodskii) claim that the advantage of the methodlies in its universality : the same set of atom-atom potentials should be useful forany molecular compound with a reasonable accuracy. Now if one varies ad libitumthe distance between the centres of interaction, the method looses its universalcharacter. Was this adjustment of the " bond length '' in N2 the only way to fit theexperimental data ?Prof.0. Schnepp (University of S. California, LA.) said: It is true that thetransferability of atom-atom potentials from molecule to molecule is impaired by thevariation of the effective interatomic distance. However, it is not clear that it shouldbe possible to represent the potential between two molecules by the interaction betweenpoint interaction centres located at the nucfear positions. After all, we are attemptingto represent the interactions between continuous distributions of interaction centres,and it is therefore not surprising that we obtain good agreement for positions of theinteraction centres different from the nuclear positions. We found in parallel workconcerning the potential models that it was not possible to fit the translational as wellas librational frequencies when the nuclear positions were used.In this case, thelibrational frequencies were consistently too high and adjustment of the positions of thecentres of interaction decreased the librational frequencies relative to the translationalfrequencies as required.Prof. J. A. Janik (Krakow) said: It seems that Brot's theory gives a possibility ofunification of dielectric measurements and neutron measurements. It might be pos-sible to obtain from a broadening of quasielastic neutron scattering line the time zD(average time between rotational jumps), and then, applying this theory to get theCole-Cole plot, which may then be compared with that obtained experimentallyfrom dielectric measurements.Dr.W. C. Hamilton (Bruokhaven Nat. Lab., N. Y.) said: We have found by calcula-tion and experiment in a number of cases that potential energy functions for rotationof molecules in solids can be extremely anharmonic. For example, one may have verybroad potential minima separated by high but narrow barriers. I wonder whetherBrot would care to comment on whether he plans to extend his calculations to otherpotentials than the one he has chosen and whether he expects that potential shapefor a given barrier height will have a significant effect on correlation functions andrelaxation times.Some of the most interesting cases of molecular rotation in crystals occur in theintermediate barrier (-kT) case where in fact there will be only a few quantumenergy levels below the barrier.With what reliability can one carry over the resultsof the classical calculation to these cases?Dr. C. Brot (Lab. de Chim. Phys., Orsay) said: The potential that we have chosenhas a sine variation W = 3 V(1- cos 0). This gives for V% kT a quasi-harmonic properlibrational frequency col = ,/2 V/I, whereas for lower V, the behaviour is anharmonicwith smaller observed frequency, as indicated in my paper. As a comparisonparabolic wells with sharp barrier, i.e., W(0) = W(n-0) = V(4e2//n2) (0<0<n/2),are broader and give a harmonic frequency w1 = (2/7c) J2 V / I . Hence the difficultyof deducing barrier heights from purely spectroscopic studies.On the other hand, reorientational relaxations studies are less sensitive to thGENERAL DISCUSSION 65shape of the potential, since this plays a role only in the configuration integral whichis present in the denominator of the pre-exponential factor (see, in the paper, theformula for pth, where the argument of the exponential should be changed for otherpotential shape), and usually the temperature variation of the reorientational rate(exponentially governed by Y ) is given more attention than its absolute value (how-ever, care should be taken not to confuse Y with the observed enthalpy of activation-see my contribution to the discussion of Dryden’s paper).We do not presently plan to examine the influence of the exact shape of eachpotential well, but rather to study systems with other symmetries than D,, e.g., of apotential with cubic symmetry and six, eight or twelve wells, which seem to be moreadequate for many plastic crystals.Concerning the VzkkTsituation for very light molecules, a first check (for fixed-axisreorientation) would be to compute the mean classical energy of the rotator and tocompare it with the quanta1 mean energies of a hindered rotator as calculated byPitzer.l It is likely that if the classical approximation is good enough for this equi-librium quantity, it would be so for the dynamical properties also, provided that thecontact with the thermal bath be good enough to ensure rapid thermalization amongthe leveis.I believe that two or three quantum levels below the barrier would thenbe enough to make the classical approximation acceptable.On the other hand,for poor contact with the thermal bath (my zi very long), tunnelling might become thedominant reorientational mechanism.Dr. M. G. Clark (University of Cambridge) said: Low-temperature anharmonicitycan be regarded as an analogue for displacement motions of the problem consideredby Martin and Walmsley. Could their approach be extended to the construction ofpotentials for the calculation of low-temperature anharmonic effects ?Prof. 0. Schnepp (University of California, L.A.) said: The lattice dynamical treat-ment proposed in the paper by Martin and Walmsley is related to the self-consistentphonon treatment of Fredkin and Werthamer and Nosanow and Werthamer andothers.This lattice dynamical treatment was devised for treating very inharmonic ornon-harmonic systems as solid helium. It was shown by Nosanow and Werthamerthat the time dependent Hartree treatment of the many body vibrational problem ofa solid can be cast in a form analogous to the classical harmonic lattice dynamicalformulation, but now instead of the potential appearing, they obtain an expectationvalue of the potential about the equilibrium distributions of the oscillators. We haverecently found that we can derive a result which is similar to that of the treatmentreferred to by using an exciton approach. We are at in agreement with the authors ofthis paper, inasmuch as librational motions of molecular solids might be successfullytreated by employing these methods.We have been successful in applying theself-consistent phonon method to the translational motions in solid hydrogen.Classical lattice dynamics using the harmonic approximation is very successful intreating harmonic systems, since for such a case the exact solution is known. On theother hand, the other methods of dealing with the many body vibrational problem,that proposed in this paper and those referred to here, have the advantage that theydo not require the development of the potential in a power series of the displacements,but on the other hand their results are not exact and represent the results of vibrationalcalculations.J. Clzem. Phys., 1942, 10, 428; 1946, 14, 239.J. G. Dash, D. P. Johnson and W. M. Visscher, Phys.Rev., 1968, 168, 1087.66 GENERAL DISCUSSIONProf. H. W. Prengle, Jr. (University of Houston, Texas) said: These comments bearon the papers presented by WaImsley and by Pawley, relating to the self-consistentfield and representation of the potential ; also to the comments of Mitra concerningi.-r. measurements as a function of pressure. We have been making measurements inour laboratory of the self-consistent field in liquids by a relatively new and uniquemethod, i.-r. shift measurements, the shift A7 =- VL-7 (gas). We use a single sharpvibrational frequency in the molecule [e.g., C=C stretch in cis-pentene-2, V(gas) =1667.3 cm-l] and observe how this frequency changes as the density of the liquid ischanged, high densities being produced by high pressure (up to 10 000 atm) and alsolow temperature along the saturation line down to the triple point and into the solid.As Elliott has pointed out, for a simple harmonic oscillation the frequency is relatedto the second derivative of the potential, and similarly for the experiment describedabove the frequency shift is a measure of the second derivative of the field potential themolecule experiences as it moves around in the liquid-and hence the self-consistentfield.The trick in making a good determination of the field is to measure Aij withsufficient precision ; since AFvalues lie in the range of 0-10 cm-' a number likeAT = 5.673 cm-l, with the uncertainty in the last significant figure, must be obtained.Shifts can be observed for the vibrator in different molecular fields produced byother non-polar and polar solvents.The results of our measurements for cis-pentene-2 down to the triple point(1 19.77 K) and into the solid have raised some interesting questions which I believeare pertinent to these discussions.The results indicate that at the same density thereis a greater shift for the solid than for the liquid. Can this be because of (i) a differentvalue of the averaged orientation function, or (ii) an additional contribution fromcoupled rotation and translation in the solid, or some other factor?Dr. R. J. Elliott (Oxford University) said: It is striking that the extra lines noted bySilvera lie at about twice the frequency of the main lines and are broader. Thisleads me to wonder whether they may be caused by two simultaneous excitations.(The terminology in this subject is a little confused.Silvera call these excitationslibrons which is all right except that the name is also used for quanta of rockingvibrations-and these are different.) There is a close analogy here to the two spinwave spectra of antiferromagnets.This explanation would be more acceptable if a suitable theoretical mechanismwere found. I do not know whether the intensities of the first-order Raman lineshave been satisfactorily explained. One possible mechanism might arise as follows.Let X he the hamiltonian of the interaction between the system and the radiationfield. The effective matrix element for Raman transitions has the formEo - Ei f ACO 'where I 0) is the ground state and [ 1) the final state of the system after scattering.I i}is any intermediate state. It must be coupled to the ground and to the final stateby the electric dipole elements of X. The incident radiation quantum is used in onematrix element and the emitted quantum in the other. In the ground state all themolecules are appropriately aligned and J, = 0 if the axis of quantization is takenalong the direction of alignment. In the final state at least one molecule, I say, has J,changed (to M,, say). However, if we apply the Frank-Condon principle X willproduce an intermediate state with an electronic dipole m but will not change M.Hence the process is forbidden unless the intermediate state is modified by a perturb-ation Y which changes f,.Then the effective matrix element will becomGENERAL DISCUSSION 67Only these values of M and m different form zero are specified in the states. Twospin wave processes can arise in a similar way with effective matrix elementNow both Vr and Vl will arise from the electrostatic interaction between the moleculeswhich may be expanded in multipoles. V, is the crystal field on molecule l arisingfrom all neighbours, V l j comes from a specific pair interaction. Both mechanismswill therefore have a similar order of magnitude, and if this mechanism is dominantwe may expect relatively intense processes with two excitations.Dr. I. F. Silvera (Science Center, California> said: Let me first try to clarify theterminology. The excitations that we call libron were originally called angularmomentum waves or librons.I believe that Homma, Okada, and Matsuda firstcalled them librons because the mean value of the rotational angular momentum Jfor the crystal is zero in the presence of an excitation. In addition, as we point outin our paper, there is a strong formal resemblance between the Q.M. and classicallibron modes,We have considered the possibility of two-libron and libron sidebands in the past.However, we did not find it necessary to include higher-order processes to explain ourlibron results in terms of our originally proposed distortion of the Pa3 space group toR3. With this structure we could account for the number of lines and their frequen-cies with a reasonable fit to theory.In addition, our calculations of the line inten-sities using the polarizability model were in fair agreement with observations. Thedifference in breadth of the high frequency and low frequency libron lines might besimply explained in terms of a two-libron relaxation process. In this case there areno two-libron states that can relax a low frequency libron and conserve energy,whereas such two-libron states exist for the high frequency lines.Our recent observation of the J = 0+2 impurity spectrum has cast doubt upon theRS space group. At the conference on quantum solids in Aspen, Colorado, H. Jamesstated that he has a preliminary theoretical result that in the neighbourhood of thetransition temperature the Cmmm space group is the most stable structure.This is atwo sub-lattice structure which would have at most four distinct libron lines. In thiscase we would have to consider the two-libron scattering processes. It would beimportant to have an estimate of the intensity of such a process, relative to the first-order process.Prof. 0. Schnepp (University of California, L A . ) said: It does not seem clear thatall observed lines in the Raman spectrum and the far infra-red spectrum of solidhydrogen must be accounted for on the basis of one-quantum processes. Two-quantum processes may be important in a system which is as non-harmonic as solidhydrogen. Since for such processes, only total q = 0 a full Brillouin zone calculationis needed, ideally including transition moments, but densities of compound states maygive sufficient information. It is difficult to do a two-phonon excitation calculation.However, calculations concerning phonon-libron combinations are promising andare in progress.Also, optical spectroscopic techniques are not the ideal tools for determination ofdetailed structure.The observations of optic phonons in cubic hydrogen in th68 GENERAL DISCUSSIONfar infra-red and the Raman spectrum presented here allow certainly the conclusionthat the structure is of lower symmetry than f.c.c. However, it is probably not to beexpected that these spectroscopic results will allow a definitive conclusion concerningthe accurate details of the structure. The experimental results presented in the paperby Silvera et aZ. represent extremely difficult measurements and the authors are to becongratulated on their successful work.Dr. I. F. Silvera (Science Center, California) said : I certainly agree that q conser-ving two-quantum processes could be contributing to the observed spectra. If sucha process is occurring in the present Raman data, then energy considerations wouldrequire it to be a two-libron process in the AJ = 0, AmJ + 0 transitions, and a libronsideband on the J = 0+2 impurity transitions. I do not believe there are anytransitions here that directly involve phonons. With the proposed R3 structure it wasnot necessary to consider two-quantum processes to account for all of the lines.However, now that this is in question we must again consider these effects.From our work we have found that many of the original theoretical ideas onordered hydrogen were qualitatively but not quantitatively correct. We now con-sider the most important experiment to be done is an accurate determination of thestructure of high purity para-deuterium in the ordered state by elastic neutron diffrac-tion. This will then give us confidence in the fundamental spectra and permit moreserious consideration of the higher order processes.Prof. J. A. Janik (Krakow) said: I suggest that we should now agree on termin-ology. My proposal is to call Zibrons, quasiparticles connected with AmJ transitions;rotons, those connected with AJ transitions and torsons, those connected with Avtransitions (torsional oscillations inside of a sinusoidal potential well)

ISSN:0366-9033    

DOI:10.1039/DF9694800061

出版商:RSC    

年代:1969

数据来源: RSC

摘要:

Lattice Dynamics of Solid Hydrofluoric AcidBY A. AXMA"*, W. BIEM, P. BORSCH, F. HO~FELD AND H. STILLERInstitut fur Festkorper- und Neutronenphysik, KernforschungsanlageJiilich (Germany)Received 1 1 th June, 1969In HF the intermolecular binding is stronger than in the other hydrohalides, presumably owing tohydrogen bonds. Because of these short-range interactions, dynamical models, which take intoaccount a limited number of neighbours only, appear better justified than for the other hydrohalides.Measurements of the energy distribution of scattered slow neutrons and the results of optical measure-ments are compared to calculations based on a model of central forces between neighbouring fluorineand hydrogen, and between neighbouring hydrogen atoms within a molecular zig-zag chain, andbetween nearest and next-nearest fluorine neighbours of adjacent chains, as well as with generalforce constants for the displacements out of the chain planes.In HF the intermolecular binding is stronger than in the other hydrohalides.Molecular aggregates form in the gas phs1se.l In contrast to the other halides, thereis no phase transition in the solid state, the molecules condense directly into a struc-ture, which corresponds to the low-temperature structure of HCl etc.2 The reasonfor this anomalous behaviour of HF probably lies in strong hydrogen bonding.Thesebonds make the internal dynamics of solid HF particularly interesting. Because oftheir short-range, dynamical models, which take into account a limited number ofneighbours only, appear a priori more realistic than for the other hydrohalides. In thefollowing results of neutron spectroscopic measurements on solid HF at 176 K arereported and compared with the results of optical measurements. Since the opticalmeasurements yield frequency distributions in the centre of the Brillouin zones and theneutron measurements frequency distributions, which are a weighted average overthe entire zone, a detailed model for the interaction forces can be derived.EXPERIMENT AND RESULTSHF was condensed into a layer of about 0.5 mm thickness and 10 x 10 cm areawithin a container of magnesium (with 3 % Al).The neutron transmission was78 %. Spectra of scattered neutrons were measured with a time-of-flight spectro-meter at the FRJ-2 reactor in Jiilich under scattering angles of 30, 55, 80, 100 and125".Fig. 1 shows typical results. The positions of the five peaks in the inelasticspectrum were independent of the scattering angle, which confirms, that the scatteringwas incoherent, i.e., due to scattering by the protons. The angular dependence of theelastic line obeyed, within the experimental accuracy, a simple Debye-Waller law.Deviations from this law could be expected for strongly anisotropic amplitudes or,more interestingly, for proton tunnelling at the hydrogen bonds, if the tunnellingfrequencies were larger than the experimental resolution and the distance betweentwo possible proton equilibrium sites larger than II/2 J K J, if K is the scattering vector* present address : Max-von-Laue Paul-Langevin Institute, Grenoble, France670 LATTICE DYNAMICS OF SOLID HYDROFLUORIC A C I D(0.8 A-l GKG2.7 A-l for o = 0 with the present experiments). The observedangular dependence of the elastic line indicates a mean thermal amplitude ,/u; =0.46 A of the proton, assuming the amplitude to be isotropic.From X-ray diffractionmeasurements the fluorine amplitudes were found to be JugII = 0.13 A in thedirection of the molecular chains and Jug, = 0.22 A perpendicular to this direction.aJneutronmeasurementsinfra-redE 10005Co Mo 50 206 10 75 5 3 2 I M OCmeWFIG. 1.-Time-of-flight spectrum of 0.005 eV neutrons scattered from solid HF.h w (meV)present* 75 27.7 16.5 10.8 7.027 7.0----__I_ =- ---- --.67 --- 58 422 379 127 98** I 68.4 45.4 25.5___--___-___-----(1 19)In table 1 the positions of the peaks in the spectrum of inelastically scatteredneutrons are compared to peak positions found in earlier neutron measurementsand in optical investigations.6-8TABLE 1INTERACTION MODELThe aim of the following calculations has been to establish an interaction model,which would (a) have as few parameters as possible, (b) fulfil the requirements ofsymmetry and stability and (c) reproduce the frequencies listed in table 1.Inprinciple, with neutron measurements a comparison could be made not only withpeak positions but with the entire intensity distribution. However, such a comparisonis straightforward only for systems of cubic symmetry;9 for all other systems,polarizations also must be calculated for all wave numbers; this has not yet beendoneA .AXMANN, w . B I E M , P . BORSCH, F . H O ~ ~ F E L D AND H. STILLER 71In the harmonic approximation, the equations of motion ares(t) being the displacement from the equilibrium position. The index a = 0,lspecifies the fluorine and the hydrogen atoms, the index i = x,y, z the components ofs, 11 the molecule in the unit cell, m the unit cell, i.e., the equilibrium positions areRP = Am+RP++f,if RP is a vector from the origin of the unit cell to the fluorine atom of molecule p.Fig. 2 shows a unit cell of the lattice : parallel zig-zag chains of HF molecules. It isuncertain, whether in adjacent chains the protons at the hydrogen bonds are at equal(fig.2) or at opposite positions. In the latter case there would be four instead of twomolecules per cell. However, for the spectrum of frequencies this does not makemFIG. 2.-The HF lattice.much difference. The calculations, therefore, were done for the arrangement shownin fig. 2. The polarizations, however, may be quite different in the two cases (seelater). For the unit cell shown in fig. 2,A = 0 2b 0 i:, li. The lattice symmetry gives the following relations among the coupling parameters(p :l72 LATTICE DYNAMICS OF SOLID HYDROFLUORIC ACIDm n n m m n hThe components c, q and 5 are indicated in fig. 2. In addition, there are the condi-tions, that no forces shall become active upon translation and rotation of the entiresystem :hh C { #;;( - 1)"(2w + v +pP)c- #" "[(u +2v)b +( - 1)"(2pP- l)d]h , O I tl :!1 h,u,B Z { # r ~ ~ ~ - ~ g ~ ( - 1 ) " ( 2 w + V + ~ P ) e = 0; (3)h hwhere d i s the distance of the fluorine atoms from the chain axis and p = Z1/l, Zl beingthe H-F, Z the F-F distance in a chain ; Rp = [0, - (- l)'d,pc] ; I?: = [O,( - 1)"2pad,pac].By trial and error a model with the following coupling parameters wasthen found to be the simplest one, which fulfils (2) and (3) and holds the lattice stable :0 (2p - 1)(B + C) +4( f4 sill2 43 + fs cos2 4,) 0000 0- fl sin2 #- fl sin # cos #- f t sin 4 cos #- fl cos2 # oQ(1) - 2 2 -'B 0 0O (fl + f 2 ) sin2 # + 2f3 sin2 9,O (fl + f 2 ) sin # cos #(fl + f 2 ) sin # cos #(fl + f 2 ) cos' # + 2f3 cos2 A .AXMANN, w. BIEM, P . BORSCH, F. H O I ~ F E L D AND H . STILLER 731 rlr = -(2p2-2p+1)(~+c) QI",'= 0 0 0 ; 2P 1 3 2p-1r12 = - [ C - ( p - l ) B ]2P 1 I0 0Q';?,' =0 f 2 sin2 # - f 2 sin 4 cos #0 f 2 sin 4 cos 4 - fz cos2 @0 0I0 f 3 sin2 9 - f 3 sin 9 cos 90 f3 sin 9 cos 9 - f3cos29sin2 4, sin @2 sin 4, -sin 43 sin 44sin qb3 sin #4 sin +2 sin #4 - sin2 #4- sin +2 sin 4, sin #3 sin #4- sin2 44sin2 4,(1;:) (Tli) (1;;) ( k y )@ oo = # pl. = 4 oo. = # oo =f4 sin@,sin+, sin24, -sin#,sin#,4 {A = $ oo = # {h = 4 hE =f4 -sin+,sin4, -sin@,sin#,i j 1 J J C j i-sin 4, sin 44 sin 42 sin #4(iii) (f'oO) (110) ( i l l )i j i j j i j i(100) (ioo) -cos241 -sin4,cos#, O6, $: = # st =fs -sin2@, O0 0 i j i j0-cos2 C#Il sin cos #1 ( i i o ) (Tio)@ $: = # $( = fs0 i j i jb-2d a* sin$, =btan = -; sin 42 =a [a2+(b-2d)2+c2]*' [a + ( b - 2d)' + c 2 ] f '. .bsin#, =[a2 +(b -2dI2 +- c 2 ] f 'The model is represented in fig. 3. The parametersf, represent central interatomicforces as indicated, f 2 represents the hydrogen bonding. The parameters B and Care general force constants for the displacements out of the chain planes. Theangles 9 and (b are indicated in fig. 2 and 3, respectively.COMPARISON WITH EXPERIMENTFig. 4 shows dispersion curves w(q) calculated for several symmetry directions withthe above parameters. A fit to the optical and the neutron data was done in foursteps : lo(l)fi was fitted to give the highest optically observed frequency at wave numberq = 074 LATTICE DYNAMICS OF SOLID HYDROFLUORIC ACID0, F; 0, H; Length of fi, 0.9281; fi, 1.5781; f3, 2.2581; f4, 3.12A; fs, 3.2081.FIG. 3.-The dynamical model.(2) The other parameters were fitted for q = 0 to the optically observed frequencies(table l), taking into account assignments of modes as done on the basis ofcomparing observations in HF and in DF.'(3) The distribution of normal states, g(o), was calculated for 756 equally distributedpoints in q-space; the parameters f4 and fs were varied such that maxima ing(m) coincided with the ones observed with the present measurements.(4) The other parameters were refitted to the optical data.15 54 ~0.2 0.4 0.6 0.8 I 0 .2 0.4 0.6 0.8 I qmax 0.2 0.4 0.60.6 IFIG. 4.--Calculated dispersion w(q)A .AXMANN, w. BIEM, P. BORSCH, F . HODFELD AND H . STILLER 75The following values were obtained (in 105 dyn cm-I) :.fi = 4.78, fs = 0.008,fi = 0.48, B = 0.220,f3 = 0.46, c = -0.121.f4 = 0.086,The large value off,, representing hydrogen bonding, is remarkable. Fig. 5 shows forthe region O<fim<120meV a comparison of the calculated g(o) to a frequency' I n'6 X5 XFIG. 5.-Above : g(w) deduced from the neutron time-of-flight spectrum as described in the text.Below : calculated g(o).spectrum deduced from the measured neutron time-of-flight spectrum (fig. 1) byconverting the time to an energy scale and by dividing through the thermal occupationfactor. As mentioned above, this evaluation prohibits a comparison of intensities ;only peak positions can be compared.In table 2 the calculated frequencies atintensity maxima for q = 0 are compared with optically observed transitions.DISCUSSIONThe model explains the neutron spectrum satisfactorily ; the intensity maxima at26 and 30 meV could not be expected to be separated in the measurements ; the twonarrow high energy peaks in the theoretical spectrum are certainly not realistic, asin this region the density of calculated points in q-space is small. The opticallyobserved transitions are also well explained, except for transition @) (table 2). Theobservation of this transition seems to have involved a number of uncertainties76 LATTICE DYNAMICS OF SOLID HYDROFLUORIC ACIDhowever : the transition with 119 meV appeared as a shoulder only ; an additionaltransition, at 98 meV, was observed, if the samples were condensed rapidly on a verycold surface.A second disagreement in the comparison of the model with the opticaldata is found in the dependence of transition (@ on M I , the hydrogen mass.TABLE 2model calculation for 4 = 0(meV) (meV)mode h W ~ ~ ~ W D FHF-antisymmetric (1) 422.1 302.1HF-symmetric (2) 378.5 273.8pseudotranslation +libration (3) 102.6 78.4libration in x-di-rection (4) 83.5 59.2libration in x-di-rection (5) 68.5 49.7libration + pseudo-translation (6) 44.6 41.8pseudotranslation +libration (7) 29.0 28.1libration +pseudo-translation (8) 27.5 20.1pseudotranslation inx-direction (9) 16.8 16.7W~~ODF-1.391.381.411.381.071.031.411 .oooptical measurements 8assigned modeHF-antisymmetricHF-symme triclibrationinactivelibration in x-directionpseudotranslation inz-directionpseudotranslation iny-directioninactiveh W ~ ~422379127 { :ds3-68.445.425.5-It seemsW~~w~~-1.351.341.381.38--1.381.030.96-probable that this disagreement is related to the large ratio f J f 5 obtained from themodel.In view of the corresponding distances Z4 = 3.12 A, Is = 3.20 A, the value10.6 forf4/f5 is surprising. It suggests, that, in contrast to the assumption made forthe present model, protons at hydrogen bonds of adjacent chains have oppositeequilibrium positions. They would then have a distance of 2.5A only, and thisasymmetric HF vibration pseudotranslation iny direction +librationsymmetric HF vibration symmetric libration +translationpseudotranslation iny direction +librattonsymrncrtrK libration in xdirectionb 1 0 v '"fk" '-*w--z 3+*19*0antisymmetric libmtion in xdirectionantisymmetric libmtion *translationtranslation in zdirectiontranslation in y-directionFIG.6.-Vibrational modes at q = 0 as derived from the dynamical modelA . AXMANN, w. BIEM, P . BORSCH, F . H O ~ F E L D AND H . STILLER 77distance would be nearly in the direction of f4. A contribution of proton-protoninteractions tof4 would also reduce the mass dependence of dispersion branch @.Three interesting observations should be mentioned (i) Fig. 4 shows that in thechain direction the branches @ and @ exhibit a strong dependence on q.This demon-strates that in the lattice the HF molecules cannot be considered rigid, in spite of thelarge frequencies of the HF vibrations. (ii) Fig. 6 represents the vibrational modesat q = 0 as obtained from the model. As indicated in table 2, most of the assignmentsdiffer from the ones made on the basis of mass dependences of observed opticalfrequencies. Those differences emphasize the value of complementing optical withneutron measurements : in systems of a complicated structure, the assignment canbe made only on complete knowledge of the lattice dynamics; such knowledge isgreatly increased with a g(w) for all q. Even more information would be obtainedby measuring, with coherent neutron scattering, directly the dispersion branches.However, with HF it seems difficult to obtain a single crystal sample.(iii) The earlierneutron measurements were done at three different temperatures, up to 280 K,where HF is liquid. The intensity peaks at 70 and at 7 meV were found at all tempera-tures. This indicates, that even in the liquid phase the chains are so stable that thelibrational motion of branch @J is little damped, and that also considerable order inthe arrangement of the chains is retained, since the frequency of 7 meV correspondsto acoustical vibrations propagating under a finite angle with respect to the chaindirection (fig. 6).The authors thank Mr. R. Joeris and Mr. A Reuters for their assistance with theexperiment and Miss L. Acker for help with the computer calculations.Gmelin, Handbuch der unorg. Chem., Erg-Bd., 1959, 5.W. Aoji and W. Lipscomb, Actu Cryst., 1954,7,173.see R. Stockmeyer, Dynamics of Orientational Defects in Adamantane, this Discussion.H. Stiller, Ber. Bunsenges, 1968, 72,94.H. Boutin, G. Safford and V. Brajovic, J . Chem. Phys., 1962, 39,3135; IneZustic Scattering ofNeutrons in Solids andLiquids, (IAEA, Vienna, 1965), vol. 11, p. 393.P. Gigukre and N. Zengin, Cm. J. Chem., 1958,36,1013.M. Sastri and D. Hornig, J . Chem. Phys., 1963,39,3497.J. Kittelberger and D. Hornig, J. Chem. Phys., 1967,46, 3099.A. Sjoelandec, Ark. F'sik., 1958, 14,315.lo P. Borsch, JUL-Bericht (Kernforschungsanlage Julich, 1969)

ISSN:0366-9033    

DOI:10.1039/DF9694800069

出版商:RSC    

年代:1969

数据来源: RSC