摘要:

GENERAL DISCUSSIONS OFTHE FARADAY SOCIETYDate190719071910191 11912191 3191319131914191419151916191619171917191719181918191 8191819191919192019201920I920192119211921192119221922192319231923I9231923192419241924192419241925192519261926192719271927SubjectOsmotic PressureHydrates in SolutionThe Constitution of WaterHigh Temperature WorkMagnetic Properties of AlloysColloids and their ViscosityThe Corrosion of Iron and SteelThe Passivity of MetalsOptical Rotary PowerThe Hardening of MetalsThe Transformation of Pure IronMethods and Appliances for 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 SIags : Their Production and Utilization in AgriculturePhysics and Chemistry of ColloidsElectrodeposition and ElectroplatingCapillarityThe Failure of Metals under Internal and Prolonged Stres?Physico-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 MolecularAtinospheric Corrosion. Second ReportThe Theory of Strong ElectrolytesCohesion and Related ProblemsLaboratoryElutriation0 rien tati onVolumeTrans.3367899910101 1121213131314141414151516161616171717171818191919191920202020202121222223232GENERAL DISCUSSIONS OF THE FARADAY SOCIETYDateI9281929192919291930193019311932193219331933193419341935193519361936193719371398193819391939194019411941194219431944194519451946194619471947194719471948194819491949194919501950I9501950195119511952195219521953I9531954t 954SubjectHomogeneous CatalysisCrystal Structure and Chemical ConstitutionAtmospheric Corrosion of Metals. Third ReportMolecular Spectra and Molecular StructureOptical Rotatory PowerColloid Science Applied to BiologyPhotochemical ProcessesThe Adsorption of Gases by SolidsThe Colloid Aspects of Textile MaterialsLiquid Crystals and Anisotropic MeltsFree RadicalsDipole MomentsColloidal ElectrolytesThe Structure of Metallic Coatings, Films and SurfacesThe Phenomena of Polymerization and CondensationDisperse Systems in Gases : Dust, Smoke and FogStructure and Molecular Forces in (a) Pure Liquids, and (b) SolutionsThe Properties and Functions of Membranes, Natural and ArtificialReaction KineticsChemical Reactions Involving SolidsLuminescenceVolume24252525262627282929303031313232333334343535 Hydrocarbon Chemistry35The Hydrogen Bond 36The Mechanism and Chemical Kinetics of Organic Reactions in LiquidThe Structure and Reactionsof RubberMolecular Weight and Molecular Weight Distribution in High Polymers.(Joint Meeting with the Plastics Group, Society of Chemical Industry) 40The Application of Infra-red Spectra to Chemical Problems 41Oxidation 42Dielectrics 42 ASwelling and Shrinking 42 BThe Electrical Double Layer (owing to the outbreak of war the meetingwas abandoned, but the papers were printed in the Transactions)The Oil-Water Interface 37Systems 37Modes of Drug Action 3938Electrode Processes Disc.1The Labile Molecule 2Surface Chemistry. (Jointly with the SociCtb de Chimie Physique atColloidal Electrolytes and SolutionsThe Interaction of Water and Porous MaterialsBordeaux.) Published by Butterworths Scientific Publications, Ltd.Trans. 43Disc.34Lipo-Proteins 6Heterogeneous Catalysis 8Physico-chemical Properties and Bebaviour of Nuclear Acids Trans. 46Spectroscopy and Molecular Structure and Optical Methods of In-vestigating Cell Structure Disc. 9Electrical Double Layer Trans. 47Hydrocarbons Disc. 10The Physical Chemistry of Process MetallurgyCrystal Growth 5Chromatographic Analysis 7The Size and Shape Factor in Colloidal SystemsRadiation Chemistry 12The Equilibrium Properties of Solutions of Non-Electrolytes11The Physical Chemistry of Proteins 13The Reactivity of Free Radicals 1415The Physical Chemist,ry of Dyeing and Tanning 16The Study of Fast Reactions 17Coagulation and Flocculation 1GENERAL DlSCUSSIONS OF T H E FARADAY SOCIETYDate Subject VolimeI9551955195619561957195719581958195919591960196019611961I9621962196319631964196419651965Microwave 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-Exci ted Species in the Gas-PhaseFundamental Processes in Radiation ChemistryChemical Reactions in the AtmosphereDislocations in SolidsThe Kinetics of Proton Transfer ProcessesIntermolecular Forces19202122232425262728293031323334353637383940For current availability of Discussionvolumes, see back cover

ISSN:0366-9033    

DOI:10.1039/DF965400X001

出版商:RSC    

年代:1965

数据来源: RSC

摘要:

tttIntermolecular ForcesBY H. C. LONGUET-HIGGINSThe University, CambridgeReceived 23rd September, 1965It is, I think, a sign of the loyalty the Faraday Society inspires, that in its 63rdyear the Society is being served by only its 3rd secretary. Many of us rememberDr. Tompkins’s predecessor, Mr. G. S. W. Marlow, who was secretary and Editorfrom 1926 until his death in 1948, but very few can have met Frederick SolomonSpiers. Spiers served the Society as Secretary from its foundation in 1903 untilhis death in 1926, and its successful development during the first 23 years, includingour world-famous Discussions, were largely due to his energy and enthusiasm.The first Spiers Memorial Lecture was delivered in 1928 by Sir Oliver Lodge, onthe occasion of our Silver Jubilee; his subject was “Some debatable problems inphysics ”.I am more than honoured to be among his successors, and to have theopportunity of opening this discussion on ‘‘ Intermolecular Forces ’,.What is a molecule, and what is a force? A molecule, for our purposes, is anatom or an aggregate of atoms sufficiently stable and permanent to be able to interactwith an environment without losing its identity. Though a clear line is difficultto draw, it is natural to distinguish between “physical ” and “chemical ” inter-actions between molecules, the former being easily reversible and the latter muchless so. In this Discussion we are mainly concerned with physical interactions.Chemical interactions usually require a close approach of the interacting partners,and the theory of short-range interactions is a branch of the theory of valency.The theory of intermolecular forces is gradually moving in to shorter and shorterrange, as later papers in this Discussion show ; but I think it is fair to say that long-range-physical-interactions are at present more completely understood thanshort-range interactions, mainly because the perturbations responsible for the formerare much smaller than those which produce the latter.Most of what I have to sayis therefore concerned with relatively long-range forces.But what is a force? For two interacting particles the answer is simple; theforce of attraction is simply the first derivative of the potential energy with respectto their distance apart.Many molecules can be regarded as rigid bodies, andthen the forces between them are derivatives of the mutual potential energy withrespect to particular relative co-ordinates. But the intermolecular potential energymay depend not only on the relative co-ordinates of the molecules but also on theirinternal co-ordinates. If this were not so, molecules could not acquire or loseinternal energy on collision with other molecules. Though we are not concernedwith inelastic collisions in this Discussion, we should not forget that their occurrencedepends on the breakdown of an approximation that we often make without ac-knowledgement, namely, that the intermolecular potential energy is independentof the intramolecular co-ordinates. This is by no means an academic point: itwould be folly to discuss the tertiary structure of proteins without recognizing thatthe van der Waals “ envelope ” of a polypeptide is determined by the angularco-ordinates within the molecule.8 INTERMOLECULAR FORCESBut even in discussing rigid molecules we often make an assumption whosevalidity is becoming more and more open to question.In many cases it seems tobe a good approximation, but sometimes it leads to incorrect expectations. I amreferring to the assumption that the mutual potential energy of three or more mole-cules is a sum of bimolecular terms, each term having the same value as it would ifall the other molecules were absent. For three inert-gas atoms at moderate dis-tances, Axilrod and Teller showed that bimolecular additivity is a good approxima-tion ; but recent theoretical work, initiated by Lifshitz and developed by Sinanoglu,McLachlan and others, suggests that in condensed phases the collective effect ofall the many-body interactions should be experimentally measureable.For systems of many molecules it is necessary to be specially careful about themeaning of the word " force ", for two different reasons. First of all, if two mole-cules A and B are near a third, C, the " force " d V / a l ? ~ ~ is undefined until one hasspecified the other two co-ordinates which are to be held constant during the differ-entiation.If U is a sum of bimolecular terms, then the natural choice is RAC andRBC ; but if U comprises a three-body term then ( ~ U / ~ R A B ) R ~ ~ , R ~ ~ is a quantityof no particular physical significance.In such a situation it is more useful toconcentrate on the energy U itself rather than its various partial derivatives.The other reason is quite different, and more important. If one is discussingthe interaction of a molecule with a surface, or of two molecules on a surface, orof two molecules in a liquid solution, one must take into account the macroscopicnature of the solid or liquid phase, and the fact that the system is usually at a finitetemperature. In principle, it might be possible to carry out an ensemble averageand obtain the mean energy of interaction between, for example, two solvent mole-cules at a given distance. In practice, however, it is much easier to make a directcalculation of the thermodynamic force, defined as (aF/dR)~~,v, where F is the totalfree energy of the solution and R is the distance between the two molecules.Thisis because the dielectric constant, which enters the expression for the force, is itselfa statistical property which can be evaluated experimentally as a function of tem-perature. The significance of these reinarks will become plainer when we come tothe later papers in this discussion; the main point 1 want to make now is that inspeaking of forces we must not confuse derivatives of U, the instantaneous potentialenergy, with derivatives of F, the Helmholtz free energy of a macroscopic system.Paradoxically, not only is F more useful than U, but its derivatives are easier tocalculate in certain types of macroscopic situation.Having reviewed the logic of intermolecular forces and, I hope, revealed someof its pitfalls, let me now say something about the physical origin of long-rangeforces and the directions in which the theory seems to be developing.2Let us concentrate our thoughts on diamagnetic molecules with closed-shellelectronic states.This description covers most of the stable molecules that wemeet in the laboratory; not all, but electronically excited molecules and moleculeswith degenerate ground states present largely unexplored problems which wouldtake us too far afield on this occasion. Suppose, then, that we have two diamagneticmolecules in their ground states, far enough apart for their electron clouds not tooverlap.In such a situation the mutual potential energy arises almost entirely fromelectrical rather than magnetic interactions between the particles of one molecuIeand those of the other. These particles are of two kinds: nuclei and electrons.The nuclei may be executing vibrational or rotational motion, but since they movH. C. LONGUET-HIGGINS 9much more slowly than the electrons it is possible to regard them as fixed, and tocalculate the intermolecular energy as a function of nuclear configuration. Theelectrons, on the other hand, are far from stationary. To a first approximationthe electron density in each molecule is determined by the nuclear configuration ofthat molecule; but in the next approximation, which we shall need, each moleculeaffects slightly the electron distribution in the other.For calculating the interaction energy between two molecules an ideal tool isquantum-mechanical perturbation theory.It was London who first showed howperturbation theory could explain long-range intermolecular forces. In a per-turbation treatment one may divide the physical effects into three categories. First,there is a direct interaction between the permanent electric moments of the twomolecules, assuming they both possess such moments. The simplest interactionof this kind is the Coulomb force between two ions; the existence of this forcewas recognized long before quantum mechanics came on the scene. Anotherexample of electrostatic forces are those which orient the molecules of a polar solventaround a dissolved ion.A third effect of the same kind is the increased cohesionwhich results from interaction between the permanent dipole moments of an as-sembly of polar molecules. One must bear in mind, however, that the interactionenergy of two permanent dipoles depends on their orientation, and that in a liquidthermal agitation tends to randomize the orientations of the molecules. For thisreason, dipole-dipole forces decrease in importance with rising temperature, so thatpolar substances often boil at temperatures not much higher than non-polar sub-stances of comparable molecular weight. The exception to this statement are almostall substances which can form hydrogen bonds. But the hydrogen bond is a subjectin itself, and we must pass on to the second kind of long-range effect, namely,polarization forces.If either of two molecules has a permanent electric moment of any order, thenits electric field can polarize the other molecule, with a consequent lowering of thetotal energy.For example, an HCl molecule will polarize a He atom, and an at-tractive force results. Or a spherical molecule at a surface may be polarized by asurface field, and be more tightly bound in consequence. On the whole, however,polarization forces seem to contribute rather little to the cohesive energies of con-densed phases, unless we are dealing with ions or with very highly polar moleculesindeed.Last, but not least, we come to dispersion forces, first interpreted by London.Dispersion forces are completely general in occurrence ; even two helium atoms,which have no permanent electric moments, will attract one another at long distances.Here classical ideas fail us completely, and we must turn to quantum mechanicsin order to understand the physical origin of these forces.To uncover the bones of the classical failure, it is instructive to consider twoone-dimensional harmonic oscillators interacting through a potential energy pro-portional to the product of their displacements.Let the unperturbed frequencyof each oscillator be VO, and let the coupling potential be ax1 x2, where a is a constant,positive or negative. As a result of the coupling, the normal frequencies v+ andv- will differ from VO, but it is not difficult to show thatv; +v2_ = 2v;.Using the expression kTlhv for the partition function of a classical oscillator wecalculate the change infree energy and find it to beAF = kT log ( V + V - / V ; ) 10 INTERMOLECULAR FORCESThis is negative because of (l), so that if I a I falls off with distance there is anattractive thermodynamic force between the oscillators.But if we analyze AFinto an energy and an entropy term we find that at all temperaturesAE = 0, AS = -k log ( v + v - / v ~ ) . (3)So not only does the thermodynamic force vanish at absolute zero, but it arisesentirely from an entropy effect !There are, of course, thermodynamic forces arising almost entirely from entropychanges; the force in a stretched rubber band is a good example. But clearly thisis not the sort of force we are looking for in considering two helium atoms.Inorder to obtain a sensible result we must quantize the energy levels of our oscil-lators, and remember that even in the ground state the coupled pair of oscillatorshas a certain zero-point energy. Doing this we obtainAE = $h(v+ + V- -2~0).Because of (1) this is negative, and if a is small it is of order a2. So althoughx1 and x2 have zero mean values for the uncoupled oscillators, the instantaneousinteraction ~ ~ 1 x 2 does lead to a second-order energy of attraction between them.The problem of interpreting dispersion forces is therefore one of discovering howthe ground state energy of a pair of molecules is modified in the second order by theinteraction between them.To prepare for the theory of dispersion forces it is helpful to begin by consider-ing the phenomenon of polarization in a little more detail.Suppose we have anon-polar molecule in an electric field, possibly the field of another molecule. Itselectron cloud will be distorted, and we may proceed to expand its electronic wavefunction in terms of the wave functions of the unperturbed molecule. The mostimportant term in this expansion will be the wave function of the ground state, butthere will be small contributions from excited state wave functions. Second-orderperturbation theory gives both the coefficients of the excited wave functions andalso a formula for the change in energy due to the perturbation. The perturbationenergy appears as a sum of small terms, one for each excited state.People some-times speak of the perturbed molecule as " undergoing virtual transitions '' to itsvarious excited states, and each transition contributes a certain amount to the per-turbation energy. If the ground state is labelled o and an excited state a, then theperturbation energy due to the virtual transition o+a iswhere Voa is the matrix element of Y between states o and a, and E, is the transitionenergy. For a molecule in a uniform electric field the numerator in (4) takes theformwhere E is the electric field and M the dipole moment of the molecule. The per-turbation energy may then be writtenW = - 4 E . a. Ewhere the polarizability tensor a is given by the relationThis equation has an interesting consequence : the polarizability of a moleculewill be greater, the lower the energy of those excited states which can be reached bH .C . LONGUET-HIGGINS 11electric dipole transitions. It reminds us of the important principle that theproperties of the ground state are strongly influenced by the presence of low-lyingexcited states.This summary discussion of polarization leads naturally into London’s theoryof dispersion forces. The essential difference between dispersion forces and polar-ization forces is that whereas the latter arise from virtual transitions involving onlythe passive partner, the virtual transitions responsible for dispersion forces involvethe simultaneous excitation of both molecules. The situation is a little more diffi-cult to appreciate intuitively because neither molecule need possess a permanentelectric field.The perturbation responsible for dispersion forces is the instantaneouscoulomb interaction between the electrons and nuclei in the two molecules. Ifthe molecules are not too close the coulomb interaction may be replaced by itsdipole-dipole component, and written aswhere the subscripts A and B refer to the two molecules and Tstands for the dipole-dipole interaction tensor. The perturbation V in this equation mixes the groundstate with states in which both molecules are excited; the second-order perturba-tion energy is given by a sum of terms of the type- I voo,ab IEoa f Eob ’voo,ab = Moa * Mob, (9)in which a refers to an excited state of A and b to an excited state of B.Thisexpression reminds one very strongly of the formula for the polarizability of anindividual molecule. It provides the basis for London’s approximate formulawhich gives the dispersion energy of two spherical molecules in terms of their polar-izabilities and ionization potentials. The ionization potentials come in becausethe most intense transitions often come near the ionization limit, so that it is notunreasonable to set every Eoa equal to IA and every Eob equal to IB in the dispersionenergy. Eqn. (10) is therefore only an approximation, but it was the first soundlybased equation to give the right order of magnitude for dispersion forces, andLondon’s theory was triumphantly successful in predicting the R-6-dependence,which follows from the R-3-dependence of the dipole-dipole interaction tensor Tin (8).London’s theory revealed for the first time the close relation between dispersionforces and the polarizabilities of molecules. Like a, W tends to be particularlylarge when a molecule has low-lying electronically excited states which can be reachedby electric dipole transitions. Thus, coloured substances often have anomalouslylarge cohesive energies, when compared with colourless substances of comparablemolecular weight. By contrast, materials such as the fluorocarbons, which aretransparent far into the ultra-violet, are often abnormally volatile ; their polariza-bilities and refractive indices are low, showing that their electrons are difficult todislodge.For many years London’s theory went virtually unchallenged, and it is a measureof his insight that we have taken so long to improve upon it.But improvement12 INTERMOLECULAR FORCESthere have been, and many of them are evident from the papers presented to thisDiscussion. Let me therefore indicate some of the directions in which our know-ledge seems to be advancing. I will concentrate on theoretical developments, sincevariousThe(9(ii)(iii)other contributors will be reviewing the experimental scene.3most important assumptions in London's theory of dispersion forces were :that the perturbation V is an instantaneous Coulomb interaction betweenthe electrons and nuclei of the two molecules ;that in the perturbation energy it is a good approximation to neglect termsof the third and higher order in V, andthat the dipole-dipole approximation for V is adequate for most purposes.Let us examine the conditions under which these three assumptions may be expectedto hold, or to break down.Three characteristic lengths enter the discussion :R, the distance between the molecules,I, the length of the larger of the two, andA, the wavelength of those photons which induce the strongest electronic transi-tions in either molecule.The easiest case to discuss is that in which the molecules are small compared withtheir distance apart (Z<R). Then the condition for the Coulomb interaction to bevirtually instantaneous is that R 4 L ; this means that a photon in the optical regionhas nearly the same phase at all points in the two molecules.For two large mole-cules lying close together (R<Z) the corresponding condition is that Z 4 A . Butif either R or Z is greater than or comparable to 1, the phase of an optical photonwill not be constant over the molecules, and one must then take account of " re-tardation effects ". This was first done by Casimir and Polder in their classic paperof 1948. For two small molecules very far apart (Z<L<R) they found that thedispersion energy falls off as R-7 rather than R-6. Assumptions (ii) and (iii) arecertainly valid in this region, and the formula for the dispersion energy iswhere aA and a3 are the static polarizabilities of the two molecules (assumed to beisotropic). This result is of great academic interest, but eqn.(11) is very difficultto check experimentally because it is only valid if the two molecules are many wave-Iengths of light apart, and then the dispersion energy is too small to be detectedby existing techniques. None the less, retardation effects do become importantfor bodies of sufficient size, and have actually been detected.London's second assumption was that one need only carry the perturbationtheory to the second order. This will be true if a typical matrix element Voo,ob issmall compared with the corresponding excitation energy &a-tE&. Now in thedipole-dipole approximation Voca,ab is the mutual energy of the transition dipolesMoo and Mob. For strong transitions these might be as large as 2 Debye; at adistance of about 3 A their interaction energy would then be roughly 0-1 eV, whileE,+Eob might be about 10 eV.For two atoms, therefore, third-order effectsshould only enter at very short distances, and one would expect them to be swampedby the short-range repulsions which arise when the electron clouds begin to overlapH . C. LONGUET-HIGGINS 13It is, however, reasonable to ask whether one might hope to observe any thirdor higher-order effects when more than two molecules are present. In the dipole-dipole approximation V is of order R-3, so that third-order effects should fall offas R-9, and one need only consider rather small values of R. The first authorsto study this problem were Axilrod and Teller. As already remarked, the second-order energy isthe third-orderthis three-bodythe sum of bimolecular terms; but Axilrod-and Teller showed thatenergy includes a three-body tern1 which varies as (RABRT~CRCA)-~ ;dispersion energy is a sum of terms of the typewhere the subscripts a, b, c refer to the molecules A, B and C .Axilrod later foundthat this three-body energy is insufficient to account for the anomalous crystalstructures of the rare gases; but later we shall be hearing that three-body forcesmay make a significant contribution to the cohesive energies of non-polar liquidsand solids. Even if many-body interactions contribute only a few percent of thecohesive energy, their presence should warn us against trying to fit the propertiesof crystals too neatly into the Procrustean bed of a two-body potential energy.I will leave till later the forces which operate within and between macroscopicassemblies of molecules, since they are best discussed from a rather different pointof view.Let me now say something about the third of London’s approximations,the dipole-dipole approximation for the perturbation V.The dipole-dipole approximation will obviously be a good one for two mole-cules which are small compared with their separation (Z4R). This is because thedipole-dipole term has a longer range than any of the other multipole interactionterms. But if I and R are of the same order of magnitude it is not immediatelyobvious how to calculate the dispersion energy. Examination shows that if onepreserves London’s first two approximations, then W will takes the formbut Voo,ab is now given byin which poa is the “ transition density at r ” for the first molecule, and pob is similarlydefined for the second.The dipole moment of the charge distribution poa is infact the transition moment M,; its higher multipole moments give corrections tothe dispersion energy which fall off more rapidly than R-6. Unfortunately, for twolong molecules lying close together the double multipole expansion of (14) con-verges very slowly if at all, and one needs to think again. Instinctively, one wouldsuppose the dispersion energy between two polymer molecules to be roughly equalto the sum of the contributions from all pairs of atoms, one in each molecule.Such an assumption is often made when people want to estimate energies of attrac-tion, but one has an obvious duty to examine it.A few years ago Dr. Salem andI investigated the problem of the “ local additivity ” of the dispersion energy oftwo large molecules, and we found that the energy will be locally additive if andonly if electron correlation dies away rapidly with distance inside each molecule.Our analysis rested on a closure approximation which is open to question, but mor14 INTERMOLECULAR FORCESrecently I have been able, by McLachlan’s techniques, to derive an alternativeexpression for W which avoids this approximation. The expression isHere ct(r1, r2, it) may be described as the mutual susceptibility at imaginary frequencyit of two points r1 and r2 in the first molecule; primed quantities refer to the secondmolecule. It measures the response of the electron density at rl to an exponentiallyincreasing potential at r2, with time-dependence exp ( t t ) .For saturated moleculesa and a’ should be small unless r1 is close to r2 and r; is close to r;. If this is so,then the four-fold space integral reduces approximately to a two-fold integral overpoints r in the first molecule and points r’ in the second. But if either moleculecontains a large conjugated system, local additivity may no longer hold and specialcare becomes necessary in estimating the dispersion energy.4I now want to talk in a little more detail about the interactions between moleculesand dielectrics, when the dielectric can be regarded as a continuum.Until a fewyears ago one had three rather different-looking theories of distant interactions :London’s theory of intermolecular forces, Casimir’s theory of retarded dispersionforces and finally Lifschitz’s theory of the forces between macroscopic bodies.Recent work, especially that of McLachlan, has resulted in a synthesis of these three,and an elegant theory has now emerged, which it seems appropriate to describeas the Susceptibility Theory.We have already noted the close analogy between London’s expression for thedispersion energy and the equation for the polarizability of a molecule; and wehave seen that in his approximate formula the static polarizability appears explicitly,McLachlan’s contribution has been to show that the relation is deeper than mightappear at first sight.I will now attempt to state, without proof, the central resultsof his theory. We first consider a pair of systems A and B for which the inter-action Hamiltonian has the special formv = -cxiyi.1Here xi and yi are certain displacements in Aand B respectively. Now to everypair of displacements in A there corresponds a frequency-dependent susceptibilityscck(w); for example, if X i and xk are components of the instantaneous electricmoment of A, then the complex quantity aik(W) will be one of the components ofthe electric polarizability at the circular frequency w. If Pik(~1)) is the correspondingpolarizability for system B, then McLachlan was able to show that the second-order interaction between A and B isummed over i and k.In this equation the susceptibility aik (it) measures the re-sponse of X i to the force conjugate to xk when this force is an exponentially in-creasing function of the time ; &(it) is obtained by analytic continuation of the functiona ( ~ ) into the complex frequency plane.The great thing about eqn. (17) is its generality. If A and B are two moleculesinteracting instantaneously through their dipole moments we obtain the Londondispersion energy. If A is a molecule and B is the electromagnetic field we obtaiH. C . LONGUET-HIGGINS 15the self-energy of A in the field. This turns out to be infinite, but one can calculatethe change in A’s self-energy caused by placing another molecule, C, in the field,and this leads to Casimir and Polder’s result for the retarded interaction betweenA and C.In passing, 1 should explain that for the electromagnetic field the sus-ceptibilities refer to two points, and describe the way in which the field fluctuatesat one point when an oscillating dipole is placed at the other. Another special caseis that in which A is a molecule and B a continuous dielectric. Then the presenceof the dielectric will modify the self-energy of A, because it changes the field sus-ceptibilities in the neighbourhood of A. Thus one obtains the energy of inter-action between the molecule and the dielectric, from the frequency dependence ofthe dielectric constant. This approach has the great advantage that one canactually measure the dielectric constant as a function of frequency for a macroscopicphase, so that there is no need to tackle a many-body problem by the laborioustechniques of infinite-order perturbation theory.Another interesting feature of the susceptibility theory is that it yields a generalformula for the second-order free energy of interaction at finite temperatures, interms of the susceptibilities at the chosen temperatures. For macroscopic phasesthese are also accessible to measurement, so again we obtain a relationship betweenexperimental quantities.The first experimental triumph of the susceptibility theorywas the well-known confirmation, by Derjaguin and Abrikosova, of Lifschitz’sexpression for the force between two solid bodies in terms of their dielectric con-stants, expressed as functions of frequency and temperature.And later in thisdiscussion we shall be hearing about yet another application of the theory, inMcLachlan’s paper on the thermodynamic force between two solute molecules ina non-electrolyte solution.Before leaving the subject of long-range forces, it may be useful to summarizesome of the main results of the susceptibility theory.(a) For two small isotropic molecules at moderate distances (Z<R<A) the dis-persion energy iswhere a(i<) aud p(ic) are the polarizabilities of the two molecules at imaginary fre-quency it. a(@ is defined as follows: if the molecule is subjected to the time-dependent electric field E exp (@), then the value of the dipole moment at time twill be or(i<)E exp (tt). @(it) is a positive monotonically decreasing function of c,and a rough-and-ready approximation for it iswhere I is the molecular ionization potential.This approximation, when used in(17), gives London’s approximate formula for the dispersion energy, already quoted.(6) For small anisotropic molecules at moderate distances the dispersion energy iswhere the subscripts are tensor indices, T5& is the dipole-dipole interaction tensor6jk 3RjRk Tjk = -- -R3 R5 16 INTERMOLECULAR FORCESand the right-hand side is to be summed over repeated indices. This expression isalso of order R-6, but depends on orientation.( c ) If the molecules are dissolved in a uniform isotropic dielectric, then anequation like (19) holds, but the various syiabols have different meanings, W isnow the free energy of interaction, and gives the thermodynamic force between themolecules. a(i5) and p(ic) are now the excess polarizabilities due to the solventmolecules, and Tjk has to be replaced by the frequency-dependent interaction tensorwhere E is the dielectric constant of the medium.(At very large distances, whereretardation becomes important, a more complicated expression for q k is required.)A rough approximation for &(it) isE(0)- 11 + (rzg/1)2’&(it)= 1 +where I is the molecular ionization potential of the solvent.approximation fails and may be replaced by( d ) For polyatomic molecules at moderate distances in vacuo the dipole-dipolein which ct and are now the ‘‘ mutual susceptibilities ” of pairs of points in thetwo molecules. This formula shows that for non-conjugated molecules the dis-persion energy is, to a good approximation, locally additive as between smallregions in the two molecules.(e) An isotropic niolecule near the surface of a dielectric is attracted to thesurface by a dispersion energy equal to(f) Two molecules reasonably far apart on the surface interact with an effectivepotential energywhich may be as little as two-thirds of the London energy.contribution to cohesive energies of condensed phases.down and is replaced by an R-7 law.small except for macroscopic bodies.energy is just the sum of the electrostatic, polarization and dispersion energies.( g ) At medium range three-body forces are small, but may make a detectable(h) At very great distances-several wavelengths of light-the R-6 law breaksBut retardation effects are undetectably( i ) In so far as second-order perturbation theory is valid, the total long-range5In the remaining few minutes I want to say something about short-range inter-actions.The most general short-range interaction is the repulsion which invariablyoccurs when two closed-shell molecules approach one another very closely. Thisrepulsion may be considered from various points of view, but in the last analysiH. C. LONGUET-HIGGINS 17short-range repulsions are due essentially to the Exclusion Principle. The reasonwhy two helium atoms repel one another, whereas two hydrogen atoms in a singletstate do not, is that there is room in the bonding orbital for only one pair of electrons ;the other pair must occupy the antibonding orbital.When account is taken of over-lap (which decreases exponentially with distance) one finds that the net effect is anti-bonding. There is, however, a slightly different point of view which makes thesituation physically clearer. One may set up a reasonable wave function for twohelium atoms by assigning two electrons to the 1s orbital of each atom. But thetotal electron density is not the superposition of contributions from the two atoms.The antisymmetry of the wave function reduces the electron density in the regionwhere the two atomic orbitals overlap. The electrons are thus forced out of theregion between the two nuclei and the net force on each becomes repulsive.Dr.Salem has shown that this picture is in good quantitative accord with the experi-mental curve obtained by Amdur from scattering experiments.The same general principles apply to polyatomic closed-shell molecules. Theatoms of the two molecules repel one another more or less independently, thoughthere may be departures from local additivity if there is extensive electron correlationin either molecule. But departures from local additivity seem to be rarer than wehave any right to expect, judging from the success of the organic chemist in anti-cipating " steric hindrance " on the basis of crude mechanical models. Further-more in some molecular crystals the work of Craig and his colleagues seems to showthat " fine adjustments " to the crystal structure are governed by repulsive forcesof a straightforward bi-atomic character.Although short-range forces between molecules are usually repulsive, this isby no means always the case, even for closed shells.If it were, chemical com-bination would be impossible and chemistry would be a very dull subject. Buteven among " physical " interactions there are two which deserve special mention,as being attractive rather than repulsive. One is the hydrogen bond, which we mustregretfully leave on one side. The other is what I should like to call the Mullikenforce, contributiug to the energy of charge-transfer complexes. In the languageof perturbation theory the binding energy of a charge-transfer complex arisespartly from a virtual transition to an excited state in which an electron has beentransferred from donor to acceptor.The real existence of the charge-transferstate is usually evident from the spectrum ; often an intense absorption band appearsin the visible spectrum, where neither partner absorbs on its own.It seems fair to say that although we understand in qualitative terms the originof short-range repulsive and attractive forces, we are still very far from a generalquantitative theory. The situation is worse still when we try to calculate three-body forces at very short distances, as I think even Professor Jansen would admit.A start has, however, been made on the forces which operate at moderate distances,when the orbitals of two molecules begin to overlap, and the overlap is still small.Murrell, Randic and Williams have made a valiant attempt to deal with this com-plicated situation by double perturbation theory, and we may expect further progressin the near future.6In a lecture of this length it has been impossible to give a balanced account ofso large a subject and I must ask you, Mr. President, to forgive me for having saidvirtually nothing about experimental problems and achievements in this lively field.But it is clear to all of us that new methods of investigation, such as the measuremen18 INTERMOLECULAR FORCESof scattering cross-sections and crystal lattice frequencies, will soon provide ampleinformation on which to test our theories. It is also to be hoped that the theoryof intermolecular forces will soon be powerful enough not only to explain the crystalstructures of simple substances but also to predict the stable conformations of themolecules which we ourselves are made of.London, 2. physik. Chem. B, 1930,11,222.Axilrod and Teller, J. G e m . Physics, 1943, 17, 349.Casimir and Polder, Physic. Rev., 1948, 73, 360.Axilrod, J. Chem. Physics, 1949, 17, 1349 ; 1951, 19, 719, 724.Lifshitz, Soviet Physics, J.E.T.P., 1956, 2, 73.Derjaguin, Abrikosova and Lifshitz, Quart. Rev., 1956, 10, 295.Longuet-Higgins, Proc. Roy. SOC. A , 1956, 235, 537.Longuet-Higgins and Salem, Proc. Roy. SOC. A , 1961, 259,433.Salem, Proc. Roy. SOC. A , , 1961, 264, 379.McLachlan, Proc. Roy. SOC. A , 1963,271, 387; 274, 80.McLachlan, MoZ. Physics, 1964, 7 , 381.Murrell, Randic and Williams, Proc. Roy. SOC. A , 1965, 284, 566

ISSN:0366-9033    

DOI:10.1039/DF9654000007

出版商:RSC    

年代:1965

数据来源: RSC

摘要:

Determination of Intermolecular Forces from MacroscopicPropertiesBY J. S. ROWLINSONDept. of Chemical Engineering and Chemical Technology,Imperial College, London, S.W.7Received 3rd June, 1965In principle, a knowledge of intermolecular forces leads directly to a knowledge of the macroscopicproperties of matter. The opposite path, that of determining the intermolecular forces from thephysical properties, is not so clear, but a few certain statements can be made. In this paper suchstatements are sought, discussed and used to test proposed two-body potentials.A knowledge of the potential energy of a group of molecules as a function of theirmutual separation and, if necessary, orientation, should suffice to determine themacroscopic properties of a substance. Equilibrium and non-equilibrium statisticalmechanics are the branches of physics that are used to elucidate these relations andboth have received a great amount of systematic study.The inverse problem, thatof determining the potential energy from the physical properties, has also been widelystudied but, on the whole, in a less systematic way. The reason is clear-there is noway of inverting the equations of statistical mechanics to give the potential as explicitfunctions of the physical properties. Hence almost all attempts to follow this secondroute have started by setting up a model potential with two or more adjustable para-meters, have calculated one or more physical properties in terms of these parametersand have concluded with a determination of the parameters by a comparison of theoryand experiment.1’ 2Much good work of this kind has been reported, and it is still our most importantsource of knowledge of intermolecular potentials.However, there is also a largeamount of indifferent work in which ill-chosen potentials have been compared withtoo few physical properties and unjustified conclusions have been drawn. This reviewis not an attempt to judge between papers in this field but its purpose is to drawattention to a number-unfortunately a small number-of definite statements thatcan be made about the potential. These results fall far short of its complete deter-mination but do serve as tests that any proposed potential must satisfy.The experimental results used are those for argon, since only an example is sought,and since these are more complete than those for any other substance.Some of thetwo-body potentials used for this substance are summarized in appendix 1.TWO-BODY POTENTIALIt is customary and convenient to assume that the potential energy of an assemblyof molecules can be written as a sum of their interactions in pairs.%(q, r2, . . . rN) = C u(rij),i < jwhere 42 is the potential energy of an arbitrary configuration and where u(rij), oruij, is the energy of two isolated molecules at a separation rij. (The dependence ofboth energies on orientation can be inserted explicitly, if required.) This division of120 INTERMOLECULAR FORCES FROM MACROSCOPIC PROPERTIESreceives some support from the quantal calculations of the various componentsof u.13 and from classical electrostatics, but neither suggests that it is exact.However,it is certainly valid in the diIute gas and so the second virial coefficient B and the low-density limits of the transport coefficients depend only on u.The second virial coefficient alone does not determine u. It is not known howmany coefficients, or indeed what physical properties are sufficient, even if (1) is valid,but probably B and almost any other one physical property should suffice.The limiting behaviour of B at low temperatures in a classical system is determinedby the shape and depth of u near its minimum. Unfortunately, this limit is difficultto approach experimentally and its relation to the potential is complicated by quantaleffects. The limit at high temperatures is determined by the positive or repulsivepart of u.If u is monotonic, i.e., if r is a single-valued function of u, then it is deter-minable 3 from B. Under tbese conditions @-1B@)] is a Laplace transform 0fr3 as afunction of u, where p = l/kT. The inversion of this transform has, apparently,not been made explicitly but seems to present no difficulty apart from that of a branchpoint at the origin p = 0. However, to integrate B over fl one must know how itbehaves near the origin and also be able to construct its hypothetical course at large pin the absence of attractive forces. Such an analysis of the measurements on heliumto 1200°C by Yntema and Schneider4 might be possible, but has not been made.If a particular form of u is assumed, say u = h - n then an explicit expression for thecoefficient A is readily obtained, but the assumption of this functional form is probablyunwise.The high-temperature limit of the transport coefficients of a gas of spherical mole-cules can also be related explicitly to u only after the assumption of a particular form.The low temperature limits are, however, more useful.They depend not on the shapeof u near its minimum but upon its magnitude at large separations. In this, theydiffer from B. They differ also in that they can usually be measured more easilyand in that the quantal corrections are less important.Their dependence on u at large separations is a consequence of their being weightedaverages of angles of deflections of colliding molecules.The lower the temperaturethe greater is the weight given to slow molecules that pass at large separations. Again,a practical use can be made of this limit only after the assumption of a particularform of u, but here one knows the form for dispersion forces, viz., u = -vr-6. (Thechange of index from -6 to -7 at very large separations is irrelevant since the tem-peratures at which these separations are important are inaccessible.) A function Aof the viscosity or thermal conductivity with the dimensions of an area, can now bedefined (see appendix 2). Its limit at low temperatures, if it can be found experi-mentally, determines v unambiguously since A3 is proportional to v. The fact thatthis area has apparently a finite non-zero limit is itself a check that the index is - 6.Fig.1 shows this function for argon. The full curves are calculated for the follow-ing potentials (see appendix l), a Lennard-Jones (12,6) potential,2 an exponential ;6potentia1,s two Kihara (12,6) potentials,69 7 and a non-analytic potential proposedby Guggenheim and McGlashan,s for which the necessary integrals have been cal-culated by Barker et al.6 The limits at T = 0 are obtained from appendix 2 and thedashed curves are interpolations. The tables of Munn et aZ.9 show that quantalcorrections are negligible for this gas above 20°K. The extrapolation of the experi-mental results to zero temperature is rather uncertain but the Lennard-Jones andexp; 6 potentials give a limiting area that is too large by a factor of at least 1-2, andhence a coefficient v that is too large by a factor of 1.7.This serious error wasunsuspected until the study of solid argon by Guggenheim and McGlashan 8 in 1960.However, their potential is seen to be irreconcilable with the viscosity and thermaJ . S. ROWLINSON 21conductivity above 100"K, and so cannot be the true two-body potential. Thecombination of quanta1 calculations and experiments on the absorption spectrum ofthe gas 10, 11 yield values of v that correspond to values of A(&') of 18.8 to 18.9 (&2,when d / k is set arbitrarily at 120°K. The experimental points are reconcilable withan area of 18-19 (A)2. Their position suggests that the Kihara potentials, althoughthe best yet tried, have too large a maximum in the area A, possibly due to theircontaining a spurious r-7 term.23 r16 1- i b---L-- _--pl._.___ I 1 w 100 150 200 250T/"KFIG.1.-The area A(€') whose low-temperature limit determines the coefficient of the dispersionpotential. The energy E'/K (see appendix 2) is chosen to be 120°K. The circles are areas calculatedfrom the viscosity 25-28 and the triangles calculated from the thermal conductivity.29-31 The linesrepresent potentials of the following types : Lennard-Jones (12,6), exponential ;6, two Kihara poten-tials (2) and (3) (KB and Kp respectively), and Guggenheim-McGlashan.Potentials of the form suggested by Kihara are the best fit to both the second virialcoefficient and viscosity. The parameters suggested by Barker, Fock and Smith 6and by Sherwood and Prausnitz 12 are close to each other and, probably, close to thetrue two-body potential.B.F.and S., n = 12, m = 6, y = 1/10, r~ = 3-363A, E/k = 142~9°K. (2)S . and P., n = 12, nz = 6, y = 1/9, r~ = 3*314A, E/k = 147.2"K. (3)THIRD VIRIAL COEFFICIENTThe most direct test of the validity of the separation of the energy (1) is providedby the third virial coefficient. If (1) is correct then the same potential u that fits thesecond virial coefficient and transport properties should give correct values of C,the third virial coefficient. If it does not, then should the discrepancy be ascribedto the failure of (1) or to a great sensitivity of C to the form of u? The latter sugges-tion has been made but not examined quantitatively.It is shown in appendix 3 thatC is, in fact, not particularly sensitive to the form of u. Small changes in u causesmall changes in C which can be estimated by the perturbation treatment proposedthere. Thus at 150"K, the critical temperature of argon and about the lowesttemperature at which C can be measured accurately, the change in C on going fromthe potential of Guggenheim and McGlashan to the '' equivalent " Lennard-Jone22 INTERMOLECULAR FORCES FROM MACROSCOPIC PROPERTIESpotential (a = 3.3 a, Elk = 90°K) with the same a andv, is only + 6 %. The observedvirial coefficient at this temperature is almost twice these values although the dis-crepancy is less at higher temperatures. This difference can be accounted for onlyby the presence of a three-body potential.The existence of such a potential was first discussed in detail by Axilrod and Teller.13The leading term in ~ 1 2 3 is proportional to (312r13323)-3 and has a coefficient which ispositive or negative according to the shape of the triangle formed by the three mole-cules.Kihara 14 estimated the size of this coefficient from the polarizability, andShenvood and Prausnitz 12 used his estimate to show that the discrepancy betweenthe two-body potential (3) and the observedvirial coefficient could plausibly be assignedto the leading term in u123.Since ~ 1 2 3 can have either sign its effect is less serious at high temperatures. Indeedit appears that both the equation of state of fluid argon compressed by shock to highdensities and temperatures, and also the course of the melting line above room tem-perature can be satisfactorily accounted for by appropriate two-body potentials only.lsPROPERTIES OF THE LIQUIDBeyond the range of the virial expansion there is no manageabIe route even fromthe intermolecular potential to the macroscopic properties, and so little that is certaincan be said about the opposite route.However, even here the form of the classicalpartition function and of certain presumed forms of the two-body potential lead tounambiguous consequences which can be used to test these presumptions. Thus,for a Kihara or Lennard-Jones potential, there is a useful identity between the potentialand its first two derivatives.16 Let u(r), the virial function, be the logarithmicderivative of u(r), and let u(r) be the logarithmic derivative of o(r).Then for thepotential (A1 .l) belowThe averages of u, v and w in a fluid are found by multiplying (4) by g(r), the radialdistribution function, and integrating over all separations. The first two averagesare thermodynamic properties, but the last is not.- = N 2 -[u(r)g(r)dr = U ,2v7 = -LF)/v(r)g(r)dr 3 2v = pv-NkT,T = -(-)Jw(r)g(r)dr, 1 N 29 2v (7)where U is the configuration energy and p the pressure of N molecules at volume F’and temperature T. However, can be found from (4) because it is sufficient to putr = a in the small terms containing y, since w(r) is large, positive and decreasing rapidlyat this separation.The following values are obtained for liquid argon at its triplepoint.16n = 12, m = 6, y = 0, 3yr = 4.33 x 104 J mole-1 (Lennard-Jones), (8)The first is unacceptable. The mean value of -W is a term 17 in the mean-squareabout T$ and the product of this fluctuation and that of @ must,-n = 12, m = 6 , y = 0.1, iy = 5.40 x 104 J mole-1 (Kihara). (9)fluctuation oJ . S . ROWLINSON 23by Schwarz’s inequality, exceed the square of the cross-fluctuation in @ and v.This inequality imposes a minimum value 18 on F, viz.,3724-49 x 104 J mole-1.If n exceeded 13 then the Lennard-Jones value of would satisfy (lo), but suchhigher values of n would not satisfy Schwarz’s inequality for the gas at high densitiesand temperatures.Kihara’s potential with y = 0.1 satisfies (10) well but disagrees with other evidenceon the value of F.Although this average is not a thermodynamic property it isrelated to div grad @ and hence to the isotopic separation factor between liquid andvapour.19 At the triple point(1 1)The only assumptions made in obtaining this value from the experimental separationfactor are that W can be expressed as a sum of pair terms and that B - 3.4 A. However,(11) differs markedly from (9). There are two possibilities. Either the Kiharapotentials (2) and (3) are not sufficiently accurate to give correctly the mean curvatureof u, and this mean is, in effect, that for separations between B and rm, or else W isnot a sum of pair terms. The former is less likely in view of the good fit of the Kiharapotential to the two-body properties (second virial coefficient and viscosity) and inview of the demonstrated importance of multi-body forces in the third virial coefficient.(10)-= 4.53 x 104 J mole-1 (experimental).PROPERTIES OF THE SOLIDThe solid can be discussed in much greater detail than the liquid but is the subjectof several later papers in this Discussion.Here all that needs to be done is to com-pare the potential that Guggenheim and McGlashan 8 “ tailored ” for solid argonwith the Kihara potentials (2) and (3). Both have about the same coefficients of Y-6and both have maximum depths of about 140°K. However, the curvature thatGuggenheim and McGlashan chose to fit the heat capacity and other high derivativesof the free energy is much less than that of the Kihara potentials and is close to thatof the best Lennard-Jones potentials.But, as fig. 1 shows, the potential cannot beclose to the true two-body potential. Conversely, (2) yields too great a lattice energy(by about 15 %) and seems unlikely to give the correct heat capacity of the crystal.Again, the most likely way out of this dilemma is to take (2) and (3) as good representa-tions of the true two-body potential, and to suppose that analysis of the solid andliquid in terms of an apparent two-body potential leads to a function which differsfrom (2) and (3) because it includes the effects of multi-body potentials.I thank Mr. J. S . Bateman of the National Engineering Laboratory, for a biblio-graphy of the transport properties of argon, and Prof.J. M. Prausnitz of the Universityof California for permission to use unpublished calculations of transport integrals.APPENDIX 1COMMONLY USED FORMS OF INTERMOLECULAR POTENTIALKihara 20 proposed the potential(Al.l)where B is the collision diameter (i.e., u(o) = 01, -8 is the minimum value of u, (yo) is a hardcore within which u is supposed to be infinite and A(m/n) is a pure number,~ ( x ) = X - ~ / ( ’ - ~ ) ( I -XI-’$ or A()) = 4, (A1 *224 INTERMOLECULAR FORCES FROM MACROSCOPIC PROPERTIESThis potential can also be written in terms of the reduced separation r/rm, where rnZ is theseparation at which u is a minimum :(A 1.3)The potential reduces to that of Lennard-Jones when y = 0. This can be written in termsof (r/a) as in (A1 .l), or asThe exponential ; 6 potential can be written in a similar form,u ( r ) = - qE (" -exp [ q ( I-- I:>1-(I;">"). q-m q(A 1.4)(A1.5)The last two potentials have the same curvature at equal minima ifn(q - m> = q(4 - m - I), (A1.6)so that a Lennard-Jones (12,6) potential is equivalent in this respect to ( A 1 3 with q = 13.77.Fig. 2 shows potentials used for argon of the type of (Al.l), (A1.4) and the non-analyticpotential of Guggenheim and McGlashan.8rlA-20 i_ --401-\I\ L .~5 6FIG. 2.-Four potential energy curves, u(Y), proposed for argon, see appendix 1 ; notation asin fig. 1.APPENDIX 2Let the dispersion potential be writtenzi(r) = - ~ ' ( r ~ / r ) ~ , (A2.1)where^' may be chosen arbitrarily, but is conveniently taken to be an energy comparable withthe expected depth of u.The distance 1-0 then determines the strength of the dispersionforces.Define a function of the viscosity q by5 = 5(mkT/n)*/16q1 (A2.2)where rn is the mass of the moleculeJ . S . ROWLINSON 25This function has the dimensions of area and for a gas of hard spheres of diameter Qis equal to &. (The so-called higher-order terms in the coefficient of viscosity areneglected.2) A similar equation can be written for the thermal conductivity of a monatomicgas. Define an area A(&') byA(&') = (kT/E')+<, (A2.3)where A depends upon the choice of 8' in (Al.1). The limiting value of A(&') as T tends tozero depends only on rg, and the numerical value of the ratio follows from the integrationsof Mason 5 and Le Fevre 21 ;A(&') = (1.194 & O.OOl)r$ (A2.4)The coefficient is obtained by a numerical integration of the scattering integral for a pure- r-6 potential. For certain mutual kinetic energies and impact parameters the trajectorypasses through the molecular centres.In general, the values of the integrals depend onwhether or not a hard core of negligible size is inserted to reflect such pairs. Its insertionwould appear to be the correct way to obtain the low temperature limit of a real potential,but the coefficients of viscosity and thermal conductivity depend only upon the square of thecosine of the angle of deflection and so are unaffected by its omission. This is not so for thecoefficient of diffusion for which the zero temperature limit seems less easily measured.APPENDIX 3The third virial coefficient has been calculated for the three potentials,21 12~22s 23 ofappendix 1.The results do not suggest that C is very sensitive to the form of u, or even tothe choice 23 of the index n, but it is useful to show that this is true in general, so that anylarge discrepancy between observed values and those calculated from u12 can be ascribedwith some confidence to ~ 1 2 3 .Consider an arbitrary potential u(r) which is not necessarily analytic at all r but whichreduces to -vr-6 at large r. Construct a potential u"(r) of a simple type (e.g., Lennard-Jones)which has the same value of v and the same collision diameter 6. Denote [u(r)-uo(r)] by6cr and assume the GulkTis small for all r>o.This is likely to be true in practice for allrealistic potentials. Nowwherefij = exp (- uij/kT) - 1.Hence C- C" = 6C is given bywherer6C = N 2 (6ul,/kT) exp (-~~~/kT)y"(r,~)d(r,-r,), J(A3.1)(A3.2)(A3.3)(A3.4)Now du is zero at large r and for r = cr. It is large for r 4 a but this is immaterial in view ofthe exponential term in (A3.3). Hence y"(r) is required only for r-rm, and in this regiononly it is known 24 as an explicit function of r and T. Hence 6C can be obtained by a singlenumerical integration. The function y"(r) does not change rapidly with r and there is nothingin the form of (A3.3) and (A3.4) to suggest 6C is particularly sensitive to the form of 6u.It will probably be a little more sensitive than 6B, for which the function equivalent to y"(r)is simply a constant, but not markedly so26 INTERMOLECULAR FORCES FROM MACROSCOPIC PROPERTIES* Fowler and Guggenheim, Statistical Thermodynamics (Cambridge, 1939).2 Hirschfelder, Curtiss and Bird, Molecular Theory of Gases and Liquids (Wiley, 1954).3 Keller and Zumino, J.Chem. Physics, 1959, 30, 1351 ; Frisch and Helfand, J. Chem. Physics,4 Yntema and Schneider, J. Chem. Physics, 1950, 18, 641.5 Mason, J. Chem. Physics, 1954, 22, 169.6 Barker, Fock and Smith, Physics Fluids, 1964, 7, 897.7 Prausnitz and O’Connell, private communication.8 Guggenheim and McGlashan, Proc. Roy. SOC. A, 1960, 225,456.9 Munn, Smith, Mason and Monchick, J. Chem. Physics, 1965, 42, 537.1oDalgarno and Kingston, Proc. Physic. SOC., 1961, 78, 607.11 Barker and Leonard, Physics Letters, 1964, 13, 127.12 Sherwood and Prausnitz, J. Chem. Physics, 1964, 41,413, 429.13 Axilrod and Teller, J. Chem. Physics, 1943, 11, 299.14 Kihara, Adu. Chem. Physics, 1958, 1, 267.15 Rowlinson, Mol. Physics, 1964,7, 349 ; 8, 107.16Rowlinson, Mol. Physics, 1965, 9, 197.17 Brown, Phil. Trans. A, 1957,250,221.18 Brown and Rowlinson, Mol. Physics, 1960, 3, 35.19 Rowlinson, Mol. Physics, 1964, 7,477.20 Kihara, Rev. Mod. Physics, 1953, 25, 831.21 Le Fevre, Con$ Thermodynamic and Transport Properties of Fluids (Inst. Mech. Eng., London,22Bergeon, J. Rech. C.N.R.S., 1958, no. 44.23 Dymond, Rigby and Smith, Nature, 1964, 204, 678.24 Rowlinson, Mol. Physics, 1963, 6, 429.25 Rietveld, van Itterbeek and van der Berg, Physica, 1953, 19, 517 ; Rietveld and van Itterbeek,26 Johnston and Grilly, J. Physic. Chem., 1942, 46, 948.27 De Rocco and Halford, J. Chem. Physics, 1958, 28, 1152.28 Flynn, Hanks, Lemaire and ROSS, J. Chem. Physics, 1963, 38, 154.29 Kannaluik and Carman, Proc. Physic. Soc., B, 1952, 65, 701.30 Ziebland and Burton, Brit. J. Appl. Physics, 1958, 9, 52.31 Keyes, Trans. Amer. SOC. Mech. Eng., 1955, 77, 1395.1960,32,269.1958), p. 124.Physica, 1956, 22, 785

ISSN:0366-9033    

DOI:10.1039/DF9654000019

出版商:RSC    

年代:1965

数据来源: RSC

摘要:

Recent Work on the Determination of thePotential FunctionBY E. A. MASON," R. J. MUNN 7 AND F. J.Received 18th June, 1965IntermolecularSMITH $The value of low-temperature transport properties and rainbow scattering for studying theattractive branch of the potential function is noted and discussed. A discussion of the use of thermaldiffusion and diffusion measurements to study the forces between unlike molecules is also presented.The calculation of intermolecular forces from quantum mechanics is possiblefor only the very simplest systems. The bulk of our quantitative informationcomes from the analysis of macroscopic data using assumed potential functions.With the advent of high-speed computers and efficient numerical techniques, it hasbecome a relatively simple matter to calculate the two-body equilibrium and non-equilibrium properties of a monatomic gas from an assumed potential function.When extracting information from macroscopic data it is essential to consider(i) which section of the potential is contributing to a particular property and (ii) ifmore meaningful information can be obtained by using a combination of properties.In the present paper it is shown that low temperature transport data and rainbowscattering experiments give specific information about the attractive branch of thepotential and that a combination of thermal diffusion and diffusion data can beused effectively to study the forces between unlike particles.LOW-TEMPERATURE TRANSPORT DATALow energy scattering is sensitive to the long-range part of the interactionpotential.The same should be true of low-temperature transport phenomena.Temperature-dependent cross-sections (" collision integrals ") for an attractivepotential, - C(6)/r6, have been calculated by Kotani.1 These integrals may be usedto calculate dispersion force coefficients as outlined by Munn.2 Thus, for a classicalgas,where A(1) and At21 are numerical factors with values 6.687 and 6.816 respectively.In fig. 1 a comparison of collision integrals for a potential, -4~&/r6, is maedwith those of a 6 : 12 potential as a function of the reduced Boyle temperature. It* University of Maryland.?&Jniversity of Bristol.$ Queen's University, Belfast.228 DETERMINATION OF INTERMOLECULAR POTENTIAL FUNCTIONis seen that transport coefficients, at temperatures below 0-4 TB, start to give usefulinformation about the magnitude of 0 6 ) .The results obtained on applying the above method to the rare gases are shownin fig.2. The meagre data available support the sum rule calculations of Dalgarnoand Kingston 3 9 4 and are approximately half those predicted by a 6 : 12 potentialfitted over a wide range of temperature to experimental data. It is obvious thatlow-temperature transport coefficients are a potentially valuable source of in-formation concerning dispersion forces.0. I 0.2TITBFIG. 1.-Comparison of transport cross-sections for a 6 : 12 potential (IRK 4) 6 : 12)) with thoseof an attractive inverse sixth power potential (QK d (6)).RAINBOW SCATTERINGWhen the interaction between two particles can be represented by a singlepotential containing a minimum, there is a critical energy below which orbitingoccurs for some impact parameter (b).Above this energy a plot of the classicaldeflection angle x as a function of impact parameter passes through a minimum(fig. 3). In classical scattering the differential scattering cross-section,becomes infinite when x equals the minimum value XRs5 Near XR, classical theoryis incorrect and Ford and Wheeler 5 have shown, using a semi-classical analysis,that the differential cross-section passes through a maximum at an angle less thanXR and through a series of subsidiary maxima and minima at smaller angles. Sincethis phenomenon is analogous to rainbows in optics, XR is called the rainbow angleand the scattering process is called rainbow scattering.A limitation of Ford and Wheeler's analysis is the small range of angles on eitherside of XR for which it is valid.Munn and Smith 6 have examined the errors in-volved and have shown the Ford and Wheeler theory gives the position of the firstmaximum and minimum correctly but not the position of the supernumerarymaxima and minima. Tables of the position of the maxima and minima as functionsof E/E and A* = h/a(2p)f have been prepared for the 6 : 12 potential. Using thh 81 L a422 0 4 06 0 8 0Ne00I IA30(P E2 octoo 200 3COA4A0O Oe @#-L I I 1too 200 300 400T/"KFIG. 2.-Calculated values of the effective dispersion forces coefficient, C(6)(T).(Units, erg cm6x1060). e, viscosity data ; A, thermal conductivity data; and 0, diffusion data30 DETERMINATION OF INTERMOLECULAR POTENTIAL FUNCTIONtables, lines can be drawn in E/e-A* space to represent possible potentials givingthe positions of the first maximum, first minimum, etc. The parameters of theassumed potential can then be deduced from the position at which the lines inter-sect. If the lines do not intersect in a point the potential model may be consideredunsatisfactory. A preliminary study suggests that the 6 : 12 potential cannotreproduce the rainbow " spectrum " found for sodium-mercury by Hundhausenand PauIy.7bluFIG. 3.-The classical angle x of deflection as a function of the reduced impact parameter, &/a,for a 6 : 12 potential with E = 4.78~.It is pertinent now to consider what information an analysis of the above typegives about the interaction potential.To investigate this it is necessary to findthe region of the potential which coiitributes most to the deflection at impactparameters close to b ~ . The classical deflection angle can be written in the formwhere g(r) = (ab/r2)( 1 - (b/r)2 - Y(r)/E)-i and r, is the classical turning-point. Infig. 4 the function g(r) is illustrated for impact parameters close to, but on both sidesof the rainbow for the 6 : 12 potential with E/& = 4-78. It is seen that the maincontribution to the deflection angle near XR comes from intermolecular distancesclose to rm. The part of the potential which contributes most to the rainbowstructure can therefore be determined by examining graphs of x against r, (fig.5).These show that for all energies the rainbow angles occur at values of r, larger than1-10, which is the position of the potential minimum. Therefore the positionsof the rainbow angles and the subsidiary maxima and minima are determined onlyby the attractive part of the potential. The actual range of the potential con-tributing for various E/E and A* have been calculated by Munn and Smith for thE. A. MASON, R. J . MUNN AND F. J . SMITH 316 : 12 potential. As an example, for EIE = 2.0 and A* = 0.01, the range of thepotential contributing to the first maximum is 1.2-1-80 and to the second maximum1.. 1 5- 1 *90.nB1.0,J b/6 = 1.30, b I6 1-46 1 2 .0 3.0riaFIG. 4.-The function y(r) for the 6: 12 potential and the same energy as used in fig. 3. Twoimpact parameters are on either side of the minimum value and the third close to it. The threevertical lines are asymptotes to the three curves and indicate the classical turning points, Y,/o.a2 00'x<- 2 0<- 4 0-661.2 1.4 1.6r m bFIG. 5.-The classical deflection angle as a function of the reduced classical turning-point32 DETERMINATION OF INTERMOLECULAR POTENTIAL FUNCTIONFORCES BETWEEN UNLIKE MOLECULESOur knowledge of the forces between unlike particles lags considerably behindthat for like particles. In many cases no direct experimental data are availableand use has to be made of combination rules to estimate the energy E and size.CTscaling parameters of the potential. Measurements on gaseous mixtures are auseful source of information on unlike forces but unfortunately for most propertiesE and CT are rather tightly coupled and variations in E can be offset by simultaneousvariations in CT.The simultaneous analysis of thermal diffusion and diffusion data enable thecombination rules for 812 to be tested independently of those for 012.The analysis of thermal diffusion and diffusion data has three separate aspects :(a) consistency of Chapman-Enskog theory and the experiments without referenceto any specific model of intermolecular forces ; (b) ability of different force modelsto fit the experimental data without reference to the combination rules; ( c ) theapplicability of the various combination rules.CONSISTENCYThe theoretical expression for the thermal diffusion factor U P contains the inter-molecular forces in the form of collision integrals many of which appear in thetheoretical expressions for the other transport coefficieats.It is thus possible toobtain a relation between CCT and other measurable quantities which depends veryweakly on the intermolecular forces (through the collision integral ratio AT2). Sucha relation can be used to check whether the measurements conform to Chapman-Enskog theory without special assumptions about forces, i.e., whether the measure-ments can be fitted by any central force niodeI.839Assuming that the thermal diffusion factor, the diffusion coefficient and theviscosity of the pure components are known, then four different consistency checksare possible: (i) the mass dependence of UT and D for isotopical mixtures ispredictable 93 10; (ii) the composition dependence of D is related to the composi-tion dependence of a r 99 11 ; (iii) the temperature dependence of D is related to themagnitude of C ~ T 8 7 9, 12-15 ; (iv) the composition dependence of CQ is related to themagnitude of D.91NTERMOLECULAR FORCE MODELSOnce the thermal diffusion and diffusion data have been found to be internallyconsistent within the framework of Chapman-Enskog theory, they can be examinedto see whether they can be fitted by some model of intermolecular forces.Thetheoretical expression for UP can be put in the formwhere XI and X;! are mole fractions.There are three factors in this equation andeach is associated primarily with one property of UT: (6C&-5) with the tem-perature dependence, the S,Q factor with the composition dependence and K12with the convergence of the Chapman-Enskog approximation scheme. Both theS,Q factor and Klz can be obtained entirely in terms of experimental quantitiesand AT,. Thus, it is possible to deduce “experimental” values of the quantitE. A. MASON, R. J . MUNN AND F. J . SMITH 33(6Cf2-5). Theoretical values of (6C&-5) depend only on the reduced temper-ature kT/t32 and thus a comparison of theoretical and calculated values of (6Cf2 - 5)gives ~ 1 2 directly. The value of 012 can then be determined from either a measure-ment of ET or D separately.The reason that (6C12-5) is especially sensitive to the details of the intermole-cular forces is shown by the following consistency formula :The derivative in the expression is usually near 2 hence (6C&-5) depends not onlyon a derivative but also on a small difference between it and a number nearly equalto it.One aspect of the analysis which still needs further attention is the behaviourof the correction term K12.The magnitude of K12 depends strongly on the specificsystem, the composition and the temperature and cannot be predicted ab initio.For instance, K12 is small for a system consisting of a trace of heavy gas plus anexcess of light gas but can be large (ca. 0.15) for a trace of light gas in a large excessof heavy gas.The calculation of K12 is laborious and has often been neglected;the results of such calculations are of doubtful significance.COMBINATION RULESA large number of combination rules have been proposed, the most commonbeingAttempts at improving the energy rule have on the whole met with indifferentsuccess. Part of the trouble is that it is hard to disentangle 612 and 812. Forinstance, Fender and Halsey 16 assumed the validity of the arithmetic mean rulefor 012 and then discussed a number of energy combination rules on the basis oftheir Ar + Kr second virial coefficient measurements. They found the best resultswere obtained with a geometric-arithmetic mean ruleE l 2 = &lE221(&11+&22).(6)But it is impossible to say if eqn. (6) is superior for ArfKr or whether it justhappened to compensate for errors in 012. This is by no means improbable; astudy of thermal diffusion and diffusion in hydrogen + krypton mixtures 9 showedthat if diffusion data alone had been studied as Fender and Halsey studied virialcoefficients, the same conclusion about eqn. (5) and (6) would have been reached.However, when both a~ and D were analyzed together it turned out that eqn. (5)was better ; the apparent superiority of eqn. (6) being due to a compensating erroron 612.On the whole, the simple combination rules seem to work surprisingly well inview of their meagre theoretical justification. Although the procedures outlinedin this section furnish an excellent route to the study of the forces between unlikeatoms, very little has yet been made of it. It would seem to be worthy of furtherexploitation.This work was snpported in part by the U.S. Office of Naval Research, under2Contract N 62558-429734 DETERMINATION OF INTERMOLECULAR POTENTIAL FUNCTION1 Kotani, Proc. Phys.-Math. SOC. Japan, 1942, 24, 76.2 Munn, J. Chem. Physics, 1965, 42, 3032.3 Dalgarno and Kingston, Proc. Physic. SOC., 1961, 78, 607.4 Kingston, Physic. Reu. A, 1964, 135, 1018.5 Ford and Wheeler, Ann. Physics, 1959, 7, 287; Mason, J. Chetzt. Physics, 1957, 26, 6676 Munn and Smith, to be published.7 Hundhausen and Pauly, 2. Naturforsch., 1964, 190, 810.8 Weissman, Saxena and Mason, Physics Fluids, 1960, 3, 510.9 Mason, Islam and Weissman, Physics Fluids, 1964, 7, 1011.10 Saxena and Mason, Mol. Physics, 1959, 2, 264, 379.11 Mason, Weissman and Wendt, Physics Fluids, 1964, 7, 174.12 Waldmann, 2. Naturforsch., 1950, 5a, 322.13 Holleran and Hulburt, J. Physic. Chem., 1952, 56, 1034.14 Holleran, J. Chem. Physics, 1955, 23, 847.15 Mason, J. Chem. Physics, 1957, 27, 782.16 Fender and Halsey, J. Chem. Physics, 1962, 36, 1881

ISSN:0366-9033    

DOI:10.1039/DF9654000027

出版商:RSC    

年代:1965

数据来源: RSC

摘要:

Extrema-Effect in Total Elastic Molecular Beam ScatteringCross-Sections for Characterization of the Potential Well*BY R. B. BERNSTEIN AND T. J. P. O’BRIENTheoretical Chemistry Institute and Chemistry Department,University of Wisconsin, Madison, Wisconsin, U.S.A.Received 9th June, 1965The theory of the extrema-effect in elastic impact spectra is reviewed and extended. It has beenshown previously that for any realistic inter-particle potential (whose well has a capacity for one ormore bound states), extrema in the total elastic molecular beam scattering cross-sections are expectedat certain characteristic velocities. The limiting high-velocity spacing of successive extrema on a110 plot is found to be inversely proportional to the product of the well depth E times the inter-particleseparation Y, at the potential minimum.The constant of proportionality is closely related to thecurvature of the well, and thus to the force constant of the di-atom (or “ complex ” molecule).Methods are discussed for the extraction of the maximum amount of information on the shape of thepotential well from measurements of the extrema-effect.It has been recognized 1 for some time that the most sensitive means of evaluatingthe intermolecular potential is the molecular beam scattering technique. Measure-ments of the differential and total elastic scattering cross-sections, da(O)/dO and CT(hereafter designated Q), over a wide range of relative velocity u are, in principle,sufficient to allow a determination of the orientation-averaged potential function V(r)over a wide range of interparticle separation r.Although formal “ inversion ”procedures (cross-sections +potential) are known,2 they have not yet seen practicalutilization. Rather, it has been customary 3 to assume a reasonable parametrizedfunctional form t for V(r) and to compute the (partial-wave) scattering phase shiftsgl(k) and cross-sections as a function of the relative velocity or the wavenumber k(=pu/fJ, where p is the reduced mass). The calculated cross-sections are thencompared with observations 4 and, by suitable variation of the potential parameters,one converges on a “ best set ” (subject to the constraint imposed by the assumedfunctionality of the potential).In general, scattering measurements at high collision energies 5 yield informationprimarily on the short-range, repulsive forces while thermal-energy 6 measurementsare especially sensitive to the long-range, attractive part of the potential.At agiven energy, differential cross-sections at the larger scattering angles (associatedwith collisions of small impact parameter 6, e.g., b 2a) are responsive to the repulsivebranch of the potential while the lower angle scattering cross-sections (governedprincipally by collisions of large b) are affected primarily by the long-range interactions.At thermal energies, due to the dominant influence of the long-range attractive“ tail ” of the potential, the angular distributions are very strongly forward peaked,so the main contribution to the total (elastic) cross-section is that from very small* This research received financial support from the National Aeronautics and Space Administra-tion, Grant NsG-275-62.t Typically the expression for V(r) involves “ strength ” parameters such as E, the depth of theattractive well, or C, the long-range (London) attractive constant, and “ size ” parameters such as 0,the “ collision diameter ” (the zero of the potential), or rm, the position of the minimum.336 MOLECULAR BEAM SCATTERING CROSS SECTIONSangles.Massey and Mohr 7 derived a simple expression for the velocity dependenceof the total cross-section (by introducing a " random-phase " approximation for themany large phase shifts of low 1 and a Jeffreys-Born (JB) approximation for the manysmall phase shifts at high I), valid for a potential asymptotically of the form V(r)- - C(6)/r6 :Q(v) = p(C'6'/hu)*, (1)where p is a dimensionless constant.*However, due to the concomitant short-range forces, undulatory deviations fromthe monotonic v-215 dependence of the total cross-section are expected, i.e., extremain the velocity dependence of Q3b99ayb.This so-called " extrema-effect'' is due to theexistence 7 of a broad maximum in the dependence of the phase shift upon the angularmomentum quantum number 1, which provides a significant number of non-randomphases at intermediate 1. The maximum phase qna increases with decreasing k and,if the attractive well is deep enough (i.e., has a capacity for one or more bound states),qm(k) can pass successively through several multiples of 7112, giving rise alternatelyto positive and negative incremental contributions A Q to the random-phase-approxi-mated Q, designated 3.The increment AQ is found to be 1*awhere Lo is the value of Z for which q attains its maximum value, qm(k); q; 3(azq/aZ2)L0. LO and 11: are also dependent upon k. Thus, the general condition foran extremum in Q is 9bwhere the integer indices N = 1,2, . . . denote maxima in Q and the half-integersN = 1.5, 2-5 . . . minima; k N is the wavenumber of the Nth extremum.For any realistic potential with a well, the high-energy behaviour of the maximumphase in the J.-B. limit has the asymptotic form$ :~ r n N c,&rmlv, (4)where the constant of proportionality c1 depends somewhat upon the particularfunctionality of the potential.For the simplest practical two-parameter potential,the L.-J. (12,6), one finds 9a c1 = 04432/h. Thus in the high-velocity limit, one hasthe following velocity dependence of the extrema :where UN is the Nth extremum-velocity. The VN may be determined by inspectionof experimental graphs of the (oscillatory) function A Q / s against u-1 (termed 9belastic impact spectra). Then, from eqn. (9, a plot of N-318 against u;1 shouldpass through the origin (thus fixing the index assignments), with the initial slope A1yielding the wm product, assuming a particular potential function with given c1.* Based on more recent treatments 8 a " best " value 8C of p = 8.08 is obtained.p This is due to the " competition " between a positive (attractive) term and a negative (repulsive)contribution to the phase shifts, characteristic of scattering by any potential with a well.In " reduced " notation, eqn.(4) may be writtenT ; - ~ l l K (4')where the " reduced phase " q* E 7/krm and the reduced collision energy K G E/E = Jpv2/=R . B . BERNSTEIN AND T . J . P . O’BRIEN 37The extrema-effect has been used (somewhat as follows) to characterize the potentialwell for a number of alkali and noble gas systems.11 First, C(6) is deduced fromQ(u) (via eqn. (1)). Then, from the elastic impact spectrum and the resulting N(u;l)plot, the w, product is evaluated, assuming a L.4. (12,6) functionality (i.e., a specificcl). Since for this potential C(6) = 2 ~ r i , E and r, may be separately obtained.These may serve as initial values in an iterative computational procedure, in whichthe potential constants as well as functionality can be altered, and the successivelycomputed A Q / ~ ( u ) curves compared with experiment.The amplitude of the oscillations in AQ/a is well approximated 10 by the semi-classical relation,-where fl = (1 + +)/krm E Z/kr, is the reduced impact parameter (Bo the value correspondingto Lo) and 0; = (dO/afl)p,, is the slope of the classical deflection function e@) at itszero, PO.Thus, given accurate experimental amplitudes one may extract the ratio Po/( -Oi)*,which is sensitive to the assumed functional form of the potential.Unfortunately,due to the blurring effect of even residual beam velocity distributions, the observedamplitude U(5) is smaller than the true U(u), so that it is difficult to improve sub-stantially on the initial value of Erm.Once the indices N are correctly assigned from a N(vG1) plot, it becomes possibleto deduce directly from the experimental extrema-velocities, via eqn.(3), absolutevalues * of the maximum phase shift qm as a function of k or v. The experimentalresults may be fitted to a power series in reciprocal velocity and ratios of successivecoefficients compared with calculations based on various assumed functional formsof V(r).The important constant of proportionality c1 (between the experimental initialslope A1 and the desired Erm product) is not fully independent of the shape of thepotential. It will be shown that (for a given E and r,) the constant is governedprimarily by the curvature of the potential at its minimum.MAXIMUM PHASE SHIFT A N D THE CURVATURE OF THE WELLWithin the framework of the semiclassical treatment of scattering, the JWKBLphase shift is independent of V(r) for r < ro, the classical turning point of the radialmotion.It is recalled 7, 9c thatJWKBL* The extrema-effect appears to be the only example of a direct experimental method for thedetermination of an absolute scattering phase shift ; ordinarily, it has been assumed that experimentcan only reveal phase shifts modulo rCT.In ref. (lob), useful tables are presented listing the three extrema-quantities -& PO and 06 asa function of K for the Kihara (12,6,a) potential with the “ core ” parameter in the range -03 <a G0.5; similar calculations for the L.J.(n,6) potential were summarized by an “ equivalence ” table,i.e., ~+XX for which the extrema-quantities were essentially identical. The experiments of Rol andRothe12 on the Li +Kr system which had been analyzed in ref. (I la) in terms of a L.-J. (124) function,were re-analyzed (lob) and a “ best fit ” of AQ/e(v) obtained with a = 0 2 , correspondingn = 23. The inference might be drawn that the repulsive index n of the interaction is deduciblefrom extrema measurements. It is suggested that a more correct interpretation of the re-analysisis that the curvature of the well is greater than that of a L.4. (12-6) well of the same e and r,,,38 MOLECULAR BEAM SCATTERING CROSS SECTIONSwhere F(r) = 1 - Y&(r)/E, ‘V&(v) = V(r) +Eb2/r2 (the effective potential), P ( r ) =1 - &/r2 and I E kb ; the outermost * zero of F(Y), i.e., F(r0) = 0, defines the turningpoint.For all impact parameters b>a (the zero of the potential), the centrifugalpotential is sufficiently great that YO >a. This follows since for Y(r0) >O,ro = b[1- V(r,)lE]-3>b>o, (8)while for V(r0) < 0, ro > o (trivially).As an illustration, fig. 1 shows the “ reduced ” classical turning points z0rro/rmplotted against fl = b/r, at various reduced energies K = E/E, for an exp (14,6)0.70 0.2 0.4 0.6 0 . 8 1.0 1.2 1-4BFIG. 1.-Reduced classical turning points for the exp. (14,6) potential as a function of reducedenergy K.The curves cross at a common point when ro = b = u, or zo = /3 = o/rm.potential. Thus, provided 7 /lo>o/r,, the maximum phase qna is independent of V(r)for all positive (repulsive) values of the potential. Strictly, then, it is not possibZeto acquire information dealing with the repulsion (V(r) > O ) from measurements of theextrema-effect. The entire information content of such experiments is confined to theattractive well. (However, from a knowledge of the shape of the well (i.e., curvature atthe minimum) one may estimate the ratio a/r, and the slope of V(r) near Y = 0.)This suggests that a suitable description of the potential from the viewpoint of theextrema analysis is one which specifies (in addition to E and r,) the curvature of the* Quantum mechanically some penetration of the wavefunction (and, for collision energies belowthe classical critical value for orbiting, tunnelling) through the centrifugal barrier occurs, but this isof secondary importance in the present connection.7 For all practical potentials this inequality is found to be satisfied; e.g., for the L.-J.(n,6)potential fi > u/rm for all n >7R . R . RERNSTETN AND T . 3 . P . O'BRIEN 39well. Thus, if V(r) is expanded around the minimum it is plausible that the leading(quadratic) term may be deducible from extrema measurements.All the semi-empirical potential functions t are readily expanded :where Y* = V/& is the reduced potential, c r z - 1 is the " reduced displacement ",z=r/rm, IC = (a2V*/a[2)o is the " reduced curvature " andf(n)(O) = (anV*/ac")o.In conventional terminology the force constant is k = ~ ~ X C ' P C O ~ = ICE/^^ = tcDe/rz,where the spectroscopic symbols have their usual meanings.One expects that inaddition to the curvature, further higher-order terms in eqn. (9) are required for accur-ate determination of q,*. (Auxiliary calculations have shown that the potential mustbe known out to z = 2 in order to achieve the requisite accuracy in q i . ) However,the principal dependence of q:(K) should be upon K ; i.e., the coefficient of the leadingterm (a1 in eqn. (4')) should be well correlated with K .The maximum reduced phase may be expanded in powers of K-4 (extendingeqn. (4')) :Nwhere the dimensionless a, coefficients are (within the framework of the semiclassicalapproximation) independent of p, E and rm, depending only on the '' shape " of thepotential.As will be shown, a1 is a slowly-varying function of K over a rather wide range of IC ;thus, &rm may be estimated to within about +lo % directly from an experimentalvalue of Al. If E and rm were known sufficiently well from independent con-siderations (e.g. to k 1 % in the &rm product), then from the experimental A1 and theknown dependence of a1 upon K one could deduce the curvature of the well (and thedi-atom force constant).COMPUTATIONAL PROCEDURESSemiclassical reduced phases q*JWKBL(fl,K) were computed via eqn.(7) for threecommonly used potentials, following procedures similar to those used earlier,l4 butwith improved accuracy of quadrature.The resulting q* values were numericallyaccurate to within about +O-OOOOl, independent of any systematic errors associatedwith the approximation (eqn. (7)) itself.The potentials were expressed in the reduced forms :potential 1 :(Lennard- Jones) v;, = (-2-)(7"Y-(A)(r:> 6 ,n-6 rpotential 2 :(11.1)(Kihara) $ ( 1-a I-a )6rlrm - a r/rm-a ' v;= ~ (11.2)7 Expansion coefficients for the L.-J. (n,6), exp (cr,6) and Morse (u)potentials have been published.13$ Slightly modified from the usual form written in terms of 6.Convergence of (9) is achieved only if O< z< 240 MOLECULAR BEAM SCATTERING CROSS SECTIONSpotential 3 :(Exponential)Vg = (L) U-6 exp[-M(c-l)]-('->(1;->6.a-6(11.3)The range of K investigated was from 1-2 to 100 (lower K would introduce thecomplication of three turning points 15 (classical orbiting), while for higher K theq: values are so small that the percentage error is larger than desired). In general,the range of p was restricted to the region in the neighbourhood of PO (e.g.,0-8</?<1-3); the maximum in the q*@) curve was obtained by a 3-point Aitkeninterpolation which also yielded Po and (Gq*lafl2),.For the eventual purpose of correlation of these results with curvature K , thefollowing equivalence relationships between potential parameters and IC wereused :potential 1:potential 2:potential 3:(12.3)Fig. 2 is a nomograph relating the various indices n, a ~ , OIE with K .(Included alsois the Morse function with index aM).K = 6n, (12.1)K = 72/(1- u K ) ~ , (12.2)K = 6 a ~ ( a ~ - 7)l(aE- 6).FIG. 2.-Nomograph relating the various indices of potentials 1, 2 and 3 with the reduced curva-ture K.The maximum phases q;(K) were computed for various values of these indicesover the range of ic from 48-108 for potentials (1)-(3). The results were expressedin terms of power series in K-* according to eqn. (10) ; the coefficients were evaluatedby a modified Aitken polynomial method. A four-term expansion fitted the valueR . B . BERNSTEIN AND T . J . P . O’BRIEN 41to within about k0.I %. It was found that, aside from the 01 (obtained theoreticallyby the J.-B. procedure), the expansion coefficients depended only very slightly on K .Ivloreover, these higher coefficients ( q a 4 ) are essentially invariant to the functionalform of the potential.Since a simple set of constant expansion coefficients (seeresults), along with the theoretical values of al, can be used to reproduce the tables ofy;(rc) to within a maximum deviation of 0-3 %, these tables will not be presented.By introducing a slight K dependence in the expansion coefficients a3 and a4 themaximum deviation can be decreased by about a factor of two.RESULTSFig. 3 shows a sample of the behaviour of q: as a function of 1/K for the threepotentials, each at three specified (common) values of curvature K. The curves areseen to group together according to curvature. The deviation from the high energy,iFIG.3.-Dependence of 7; upon 1/K for potentials 1,2 and 3, each at the specified values of K.limiting behaviour (eqn. (4’)) is best shown in graphs of the product q$K against l/K,as illustrated in fig. 4 for the same set of calculations. Here again the “ family ”structure is apparent, curvature being the dominant factor. The ordinate interceptTABLE 1K4860728496108120132144.-EXPANSION COEFFICIENTS (a1-a4) FOR q:(K) ACCORDING TO EQN. (10)a1potential 1 putential 2 potential 3a47000 -47421 -45209-44113 -44432 -43063-421 56 -421 56 -41462-40727 -40338 -40232-39630 038838 -39257-38757 -37568 -3 8465-38043 s364-74 -37808a37446 e35515 -37253-3 6938 a34666 -36776a2= 2-Oxlo-3; a3= -1*90x10-1; @=I 8-8x10-242 MOLECULAR BEAM SCATTERING CROSS SECTIONSof the curve (via extrapolation) gives a " computationally evaluated " al.Theseagree well with the J.-B.-calculated a1 values.The constants of the semi-empirical equations for qz (following the form of eqn.(10)) are presented in table 1.I I_-_c_ ~1 lKi 0250 d.1 0.2 d.3 0.4 66 06 0 7 06 WJ MFIG. 4.-Dependence of r]zK upon 1/K for potentials 1, 2 and 3, as in fig. 3.M2' I ' I I ' ' ' ' ' 50 I 0 0 150KFIG. 5.-Dependence of the u1 parameter upon K for potentials 1, 2 and 3. The expressions fora l ( ~ ) in the three cases are given in the appendix.The a1 coefficients listed are the theoretical (3.-8.) values, while the higher co-efficients were evaluated by the Aitken fitting procedure. The al's were expressedin closed form only for potential 1.The analysis for the other two potentials involvedthe solution of transcendental equations and power series expansions (see appendix)R . B . BERNSTEIN AND T. J. P . O'BRIEN 43The K-dependence of the important parameter a1 is shown in fig. 5. Here al(rc) showsa broad region (70<~<130) in which (i) the slope is essentially constant (dal/dlc z- 9 x lO-4), and (ii) the results for the three potentials are very similar.CONCLUSIONSThe excellence of the correlation of a1 with K, nearly independent of the particularfunctionality of the potential, confirms the suggestion that the third " characterizationparameter" of the potential should be the curvature of the well. Since the higherorder coefficients (a2-a4) are found to be completely invariant to the functionality ofthe potential and essentially independent of K, an experimental determination of theratios a4al and a3/al (from plots of NVN against vil) should allow (via table 1) anestimate of K and thus the di-atom force constant.The authors appreciate the assistance of Mrs.N. Gordon and Mr. T. T. Warnockin connection with certain aspects of computing and plotting the results. Inaddition, one of us (R. B. B.) wishes to acknowledge some valuable discussions on thesubject of this paper with Dr. E. W. Rothe.APPENDIXHigh-energy limiting forms for the extrema-quantities q; and PO may be obtained fromthe J.-B. approximation for the phase shift.9.10fls 16 These were obtained in closed formonly for potential 1. In this case,where 16For potential 2,The integrals 17 were evaluated by expanding the factor of (1 - OI/.X)-" in a convergent powerseries.Then q* can then be written in the form(k+6)1 ak+-t 120(k+ l)!, (ki-5)! [(T)!]' (1 -@ --2(1 -a)62--( k + ll)!ak11 ! k ! 1 - 3 . . . (k+lO)fik+"1 ( k + l l ) ! [(?)!-J (A-544 MOLECULAR BEAM SCATTERING CROSS SECTIONSSubstitution of PO (the solution of the equation (aq*/aP = 0) in eqn. (A-5) yields qgJB.Sufficient terms were carried in the expansion to insure convergence of the value of qmJB tof0-000001. This required terms as high as k = 40 for low K (i.e., negative a).To evaluate the corresponding quantities for potential 3 to the requisite accuracy, thenecessary higher order expansion terms were added to eqn.(lob) of ref. (16). This equationthen becomes- . . .]}. (A-6)15 105128a2P2 i- 1O24a3p3-Again qZm may be evaluated by substituting PO (found by differentiation of q*IB) in eqn. (A-6).It can be seen from eqn. (4’) that the expressions for a1 are obtained directly from eqn.(A-l,5,6) by suppressing the factor 1/K.for example, (a) Massey and Buckingham, Nature 1936, 138, 77; for recent general review,see ( b ) : Bernstein, Science, 1964,144, 141.2 Hylleraas, Physica Mathematica Univ. Oslo Rept. no 19 (1963); see also (b) Faddeyev, J. Math.Physics, 1963, 4, 72.3 see, for example, (a) Bernstein, J. Chem. Physics, 1960, 33, 795; (b) ibid., 1961, 34, 361 ;(c) Mueller and Marchi, ibid., 1963, 38, 745 ; (d) Brackett, Mueller and Sanders, ibid., 1963,39, 2564 ; (e) Sanders and Mueller, ibid., 1963,39,2572.4 for a recent example of this procedure, see Groblicki and Bernstein, J.Chem. Physics, 1965,42, 2295.5 (a) Amdur and Pearlman, J. Chem. Physics, 1940, 8 , 7 ; the first of an extensive series on highenergy molecular beam scattering. For review see: (b) chapter by Amdur and Jordan inMolecular Beams, ed. by J. Ross (vol. 10 in Ado. Chem. Physics, series, (Prigogine, ed.), (J. WiIeyand Sons (Interscience Div.) New York, 1965).6 for review, see Bernstein in Atomic Collision Processes, p. 895, (ed. McDowell), (North-HollandPubl. Co. Amsterdam, 1964).7 Massey and Mohr, Proc. Roy. SOC. A, 1934,144,188.8 (a) Schiff, Physic. Rev., 1956,103,443 ; (b) Landau and Lifshitz, Quantum Mechanics, (Pergamon,London, 1959) ; (c) Bernstein and Kramer, J. Chem. Physics, 1963,38,2507.9 (a) Bernstein, J. Chem. Physics, 1962, 37, 1880; (b) ibid., 1963, 38, 2599. See (c) chapter byBernstein in Molecular Beams, (ed. Ross) (vol. 10 in Adv. Chem. Physics series, (Prigogine, ed),(J. Wiley and Sons (Interscience Div.) New York, 1965)).10 (a) Diiren and Pauly, Z. Physik, 1963,175,227 ; (b) ibid., 1964,177,146.11 Rothe, Rol and Bernstein, Physic. Rev., 1963, 130, 2333 ; (b) Rothe, Neynaber, Scott, Trujilloand Rol, J. Chem. Physics, 1963,39,493 ; (c) Duren and Pauly (see ref. (10)) ; ( d ) Von Busch,Strunck and Schlier, Phys. Letts., 1965, 16, 268; (e) Herschbach and co-workers (privatecommunication, May 1965); (f) Pauly and co-workers (private communication, Sept. 1965);(9) for a summary of expt. results see Pauly and Toennies, Adv. Atomic and MoZ. Physcis(ed. Bates and Estermann) (Academic Press, N.Y., 1965) vol. 1, p. 302.12 Rol and Rothe, Physic. Rev. Letters, 1962, 9, 494.13 Harrison and Bernstein, J. Chem. Physics, 1963, 38, 2135.14 Weber and Bernstein, J. Chem. Physics, 1965, 42, 2166.15 see, e.g., Curtiss, J. Chem. Physics, 1965, 42, 2267.16Bernstein, J. Chem. Physics, 1963, 38, 515.17 Bernstein (chap. 3 ) in Molecular Beams, (Ross, ed.), vol. 10 of Ado. Chem. Physics, (Prigogine,ed.), (J. WiIey (Interscience) N.Y., 1965), eqn. (lVb,2a)

ISSN:0366-9033    

DOI:10.1039/DF9654000035

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年代:1965

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Inversion Problem in Molecular Beam Determinations of theIntermolecular PotentialBY JOHN LUOMA AND C. R. MUELLERChemistry Dept., Purdue University, Lafayette, IndianaReceiGed 1st June, 1965Since the molecular beam technique appears to be very sensitive to the details of intermolecularforces, an iterative method has been developed to obtain the phase shift curve from differential cross-sections, followed by recovery of the potential from phase shifts. The primary problems whichhave arisen were those of convergence and uniqueness. One must use a reasonably accurate startingpoint and scattering data out to the second maximum has also been found to be necessary. Themodification of the procedure for experiments without velocity selection has been considered.The molecular beam technique appears to be the most sensitive method of deter-mining the intermolecular potential in the attractive region.1 We shall be primarilyconcerned with the use of differential scattering cross-section data for this purpose,and most of our attention will be directed towards low-angle scattering.1.SENSITIVITY OF THE DIFFERENTIAL CROSS-SECTION TO VARIOUSFEATURES OF THE INTERMOLECULAR POTENTIALIn general, we find that small-angle scattering is characterized by the presence ofa maxima at zero degrees and with a succession of maxima at higher angles. Wehave chosen, for extensive study, the region between the first and second maxima.The extent of this region varies from 15" (He+Hg) to a few minutes of arc. Thehigher figure was found to be more normal for rare gas collisions at thermal velocities.This region accounts for almost all of the total scattering cross-section and thereappears to be some confusion as to the features of the potential which are responsiblefor its magnitude.It has been widely held that small-angle scattering is due to long-range forces.2-5 This idea has lead to the use of total scattering cross-sectionsand small-angle scattering for calculating long-range force constants.We have investigated two systems which are more or less representative of molecularscattering. The first of these was for He+Hg collisions.6 We found, in this case,that the curvature of the region near the potential minimum had a large effect on thedifferential cross-section between 0 and 5".The phase-shift curve for this systemhad a large maximum near three radians.The second system He+Ar had a phase shift curve which had a maximum at0.6 radians. We have made a series of calculations holding the long-range forcesconstant. Fig. 1 shows three potentials which have their minima at different positions.The effect of this shift in fig. 2 is small below 5" but is very large beyond 8".In fig. 3, two potentials are shown in which the forward curvature of the bowl hasbeen altered. In fig. 4, the change in the resultant differential cross-sections is shown.There are large changes near all of the maxima with small changes near the minima.Varying the position of the bowl changed the total cross-section from 102 to 114 A2.Varying the curvature of the bowl caused a change from 102 to 125 &.The differen-tial cross-sections were thus found to be much more sensitive to the potential thanthe total cross-sections.4.46 MOLECULAR BEAM INVERSIONQCh r- €I3.0 4.0 5.0FIG. 1 .-Three potential energy curves representing He+ Ar. The long-range potential wasidentical in all three cases as was ~(294 x 10-15 ergs).hI IO0FIG. 2.-Differential scattering patterns for He+Ar illustrating the effects of the potential varia-tions of fig. 1. The line symbols have the same meaning as those of fig. 1J . LUOMA AND C . R . MUELLERr __-_47FIG. as0" 5" 10"6C.M. (deg-1FIG. 4.-Differential scattering patterns for He+ Ar illustrating effects of potential variations offig.3. Line symbols are identical to those of that figure48 MOLECULAR BEAM INVERSIONThe primary effect of the long-range forces was found to be in the small-angleregion, below 3" for He+Hg,6 and most important at 0". There was a roughcorrelation between the region of the potential and the angular region which it primarilyaffected. The long-range forces had their major effect at very low angles and theirlargest effect at 0".The outer portion of the potential well had an effect distributed over the entiresmall-angle region. The inner portion of the well had its greatest effect beyond 8".There was extensive overlapping, however, and these results strongly suggest the needfor an inversion programme of the type which is described in the next section.2.INVERSION OF VELOCITY-SELECTED DATAOur experience with curves of the types in fig. 1-4 led us to suppose that all of thevarious parameters of the potential must be determined simultaneously from thescattering curve by an inversion technique. Our initial attention was directed towardthe kind of data that one would get from velocity-selection.The technique which we have used 7 involves expressing the phase shifts as afunction of parameters which give rise to a smooth curve :where 61 is the phase shift, and 2 is the angular momentum quantum number. Onlyfive of these parameters have been varied at one time. Since 61 was expressed as afunction of several parameters, it might seem logical to extend this to its ultimatelimit, and use as the parameters the vaIue of Sl at each 2-value. We have tried thisand we could reproduce scattering data which was generated from a smooth phase-shift curve, but the phase-shift curve had an erratic variation of phase-shifts forconsecutive I-values.The functional relationship in eqn. (2.1) was used to provide areasonably smooth phase-shift curve. This can be regarded as a physical limitationarising from the behaviour of a smooth potential.We have dealt with the problem of uniqueness from a purely pragmatic standpoint.The information which we can expect from experiment will be limited in accuracyand restricted to certain angular regions. We have rough ideas as to the behaviourof the intermolecular potential. The question of uniqueness, as we have investigatedit, was treated according to the criteria of the reproducibility of a phase-shift curvefrom the data using a range of starting points.We have started with an estimate of the potential and with the scattering data.The potential provides us with a phase-shift curve which is expressed in terms of eqn.(2.1).The differential cross-section was expanded about the initial estimates of theparameters C k by a Taylor series expansion. New values of the parameters werechosen which minimize the difference between observed and calculated cross-sections.A Taylor series expansion was done around the new values of the parameters and theprocess was repeated until convergence was obtained.We have found that the large negative phase-shifts were unimportant in determin-ing small-angle cross-sections.7 These phase shifts are weighted by small 2-values and,since they pass rapidly through multiples of n, tend to give random contributions.There is good reason to suppose that they are better determined by transport pro-perties.1 The large negative phase-shifts were usually held constant during an itera-tive cycle.The other regions of the phase-shift curve were sensitive to the scatteringpattern.If the differential scattering data was not over a sufficiently wide region it wouldnot determine a unique phase-shift curve. Convergence to the scattering data woulJ . LUOMA AND C. R. MUELLER 49occur but different phase-shift curves would result for He+Hg, when the data coveredthe first 9 degrees. Data up to 15 degrees was required.One of the annoying features of this programme was the tendency for one of theparameters C4 to oscillate wildly.If an initial estimate was particularly bad therewas frequently a rather large overcorrection. Sometimes the overshoot would buildup and an excursion occurred outside the range of the computer. We have ex-perienced either convergence or an excursion which stops the computer.Table 1 illustrates the kind of convergence that one can expect from this procedure.The “data” was calculated from a Lennard-Jones potential intended to fit theHe + Hg data (E* = 1.54 x 10-14 ergs, CT = 2.35 &v = 1.0 x 105 cm/sec). The functionused in eqn. (2.1) still lacks enough flexibility to fit the data precisely near the firstminimum.The results of the tenth cycle were almost identical to those of the fifthcycle.TABLE 1 .-CONVERGENCE OF DIFFERENTIAL CROSS-SECTIONSdo/&(&) x 10-4angle,13579111315data0.4170.2180.07790.02190.00580.00680.0 1070,0087irput0.5850.3320.1 160.0260-00950.01460.0 1470.0084third cycle0.4000.2070-07430.02000.00620.00680-00860.0086fifth cycle0.4160.2170-07850.02140.00610.00740.00950.0087Most of our effort has been concentrated on the phase-shift problem. Theintermolecular potentials can be obtained from the phase shifts in a similar way.8The potential was expressed in terms of a number of parameters. A Taylor seriesexpansion was used to find a function which described the behaviour of the phaseshifts in the region of the initial potential estimate.Enough points on the phase-shift curve were used so that the potential parameter could be determined. Theprocess was recycled until no further improvement resulted. The final potentialwhich was obtained depended to some extent on the points of the phase-shift curvewhich were used. The best results were obtained from points spread over a widerange, (5 % error in depth, good fit otherwise). Probably these parameters need tobe overdetermined using a least squares fit.To summarize, a method for the inversion of velocity selected data has beendeveloped. While many improvements need to be made in fitting functions andprocedure, the bare outlines of a practical method are present.In order for the inversion technique, which we described, to be technically possible,it must be extended to experiments without velocity selection.In the scattering ofhelium by mercury a velocity spread of 5 % would probably be tolerable, and this iswithin the reach of present techniques. Some data obtained by Bernstein and co-workers on Li+Hg comes close to this criterion,g. 10 but the cross-sections were notabsolute measurements.For encounters between molecules of similar mass, double velocity selection isrequired as well as a sharply-defined scattering beam. Present crossed-beam experi-ments use very closely-spaced wide slits to produce a scattering beam intense enoughto provide appreciable attenuation. It appears that formidable problems stand inthe way of velocity selection for collisions of any but the lightest molecules with veryheavy molecules50 MOLECULAR BEAM INVERSION3. INVERSION OF SCATTERING DATA WITH VELOCITY DISTRIBUTIONSFor the present, one must deal with the problem of analyzing data from Maxwellianbeams. The simplest way to extend the inversion technique was to find a way ofgenerating a velocity averaged cross-section from a monochromatic scattering curve.This required a study of the effect of velocity on both the angular behaviour andmagnitude of the differential cross-section.If we could find such relationships,the inversion technique for unselected data would closely follow the iterative techniquepreviously developed.4.c3 .OI I I 10" 6" 12" 18"0C.M.(deg.)FIG. 5.-Transformed scattering patterns for He+Hg calculated at thee relative velocities ;v = 1.375 x 105 cm/sec solid line ; u = 1.625 x 105 cm/sec circled line ; v = 2-00 x 105 cm/secdashed line. The L.-J. (12,6) potential was employed with E = 1.54 x 10-14 erg and Y* = 2.64 A.The values of 0C.M. are those of the middle velocity and eqn. (3.1) was used to transform the scatteringpatterns of the other velocities.The simplest relationship was found for the dependence of extrema of the differentialcross-section upon the velocity. Bernstein 10 has found an approximately linearrelationship between the angular positions of the extrema for a Lennard-Jonespotential. In fig. 5 we have illustrated this linearity by using eqn.(3.1)8' = (OV)/U' (3.1)to transform angular dependence of the differential cross-sections at various velocitiesto the scale of the central velocity. While this relationship was not exact, correctionswere made empirically at later stages. An examination of fig. 5 reveals that thJ . LUOMA AND C . R . MUELLER 51intensity relationships are complicated. At zero degrees, for helium + mercury,6the cross-section increased with increasing velocity. At angles higher than 6' >20°,the reverse was true. A closer examination of the zero angle cross-section 6 hasshown a sigmoidal dependence which could be only represented by a cubic, or higherorder, equation.Since eqn. (3.1) is not exact, and since the intensity relationships are complicatedwe have used an empirical procedure which is cumbersome but probably necessary.We have used a trial potential to calculate a velocity-averaged differential cross-section.The monochromatic cross-sections were used to establish an angularrelationship,where b was small. A central velocity was selected and eqn. (3.2) was used to trans-form from a given angle to an equivalent angle at another velocity. This point wasgiven a weight g which depends on intensity and statistics. Since the former is atleast cubic, the form,0 = (avt/v+b>6', (3.2)g = CCnvn exp (- OIU'), (3.3)suggests itself. We used terms for n = 4,5 and 6 for He + Hg.FIG. 6.-Comparison of least-squares averag-ing procedure indicated by + with Knauer's 11data (circles) and Marchi's 6 velocity averag-ing procedure shown by solid line.Eqn. (3.2)was used in averaging over 0c.M. and eqn.(3.3) was employed in averaging over differen-tial scattering intensities.The constants were determined by a least-square fit of a velocity average computedby Marclii.6 The results are shown in fig. 6. Originally Marchi used eleven velocities,and the curve possessed several undulations. This was in contrast with H2+Hgwhere eighteen velocities were used and a smoother curve obtained. The angulardependence on velocity was a primary factor in the smoothing process.This procedure was used in analyzing the data of Knauer 11 on scattering of Heby Hg and the phase shift curve has been obtained while obtaining a good fit, wellwithin the limits of probable experimental error for all of the data between 0 and17".The changes in the phase shift curve were very complex. At high I-values52 MOLECULAR BEAM INVERSIONabove 40, the phase shifts were larger than those from the best Lennard-Jonespotential used by Marchi.6 This indicates somewhat stronger long-range forces.In addition, the phase shift curve had a maximum which is shifted to lower I-values(to 18 from 26) indicating a minimum in the potential at smaller interatomic separa-tion. Finally the maximum of the phase shift curve is much higher (to 2-6 radiansfrom 1-8 radians). This is a good indication of a broadened potential, but thismay be coupled with a change in the depth of the potential.At this stage in the iteration, the inversion of the phase shift curve has been re-duced to the monochromatic problem which has been previously discussed8 andwe are presently in the process of obtaining the first approximation to the inter-molecular potential.This research was supported by the Advanced Research Project Agency.1 Mueller and Brackett, J. Physics, 1965, 40, 654.2 Massey and Mohr, Proc. Roy. SOC. A , 1934,144, 188.3 Massey and Buckingham, Nature, 1936, 138, 77.4 Rothe and Bernstein, J. Chem. Physics, 1959, 31, 1619.5 Pauly, Fort. der Physik, 1961, 9, 613.6 Mueller and Marchi, J. Chem. Physics, 1963, 38, 745.7 Brackett, Mueller and Sanders, J. Chem. Physics, 1963, 39, 2564.8 Sanders and Mueller, J. Clzem. Physics, 1963, 39, 2572.9 HostettIer and Bernstein, Physic. Rev. Letters, 1960, 5, 3i8.1 0 Groblicki and Bernstein, J. Chem. Physics, 1965, 42, 2295.11 Knauer, 2. Physik, 1934, 90, 559

ISSN:0366-9033    

DOI:10.1039/DF9654000045

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年代:1965

数据来源: RSC

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GENERAL DISCUSSIONProf. E. A. Guggenheim (Reading University) said : Rowlinson raises the questionhow rnafiy physical properties are required to determine the interaction energy curveand he surmises that two may be enough. I think he is optimistic, but the degree ofhis optimism depends on his precise meaning. If he is concerned only with theheight of the curve, but not with its curvature nor other details of shape, he may beright. In this restricted sense, of the four curves in his fig. 2 the one labelled L.J.is wrong while the other three are right.If, however, we are interested in the detailed shape of the curve then each ofMcGlashan’s parameters has a clear physical significance. There exists a temperatureat which the equilibrium distance between nearest neighbours is precisely equal to YO.At this temperature, while ro determines the lattice constant, K determines the entropyand E determines the energy. The temperature dependence of these quantities isdetermined mainly by a.The contributions of anharmonicity are determinedmainly by p/a - 1. The interaction between non-nearest neighbours is determined by1. To fit the viscosity at least two further parameters will be required to describethe steep repulsive part of the curve. Thus at least eight parameters are required.I doubt the possibility of fitting the whole curve by a single formula. If this is possiblethen the number of parameters required will be still greater.Prof. J. S. Rowlinson (Imperial College, London) said : Prof. Guggenheim and I agreethat the Lennard-Jones (12,6) potential is not an adequate representation of the pairpotential of argon and that, in particular, it gives too high a coefficient of r-6.Weappear to disagree over the probable importance of the three-body forces. There canbe no certainty here yet, but I believe that the body of the evidence is that these forcescannot be ignored.My first reason is that the observed peak in the third virial coefficient is too largeto be accounted for by any pair potential which also fits the second virial coefficientand the transport properties. Sherwood and Prausnitz showed that the size of thispeak can be convincingly accounted for by a dipole-dipole-dipole term with acoefficient related to that of Y-6. Barker has confirmed the accuracy of that relationat this Discussion.My second reason is that the effective pair potential for the solidthat Guggenheim and McGlashan proposed in 1960 is not reconcilable with theviscosity. I agree that Barker’s calculation of the collision integrals for this potentialwas made with the rather arbitrary collision diameter of the 1960 paper, and that theassumption of a more realistic continuation to short separations might improve theagreement.When the best pair potentials for the gas are used to calculate the properties of thecrystal then they predict too small a latent heat of sublimation and have a curvaturenear the minimum which exceeds that of the potential of Guggenheim and McGlashan(and of the revised potential proposed below by McGlashan). The first discrepancycan probably be accounted for by the three-body potential, as Rossi and Danon showbelow, and the discussion of McGlashan’s paper suggests strongly that his estimateof the curvature K is too low even for the effective potential of the crystal.Anyremaining discrepancy here, and the similar discrepancy in in the liquid, might bealso a consequence of the three-body potential, but no direct test of this has yet beenmade.Dr. G. Casanova (Universitd di Genovd) said: A different kind of analysis ofexperimental data on argon for the isotopic separation factor between liquid and53However, there is no evidence that this is so54 GENERAL DISCUSSIONvapour is possible through the use of the radial distribution function g(rj.1 Thoughthis approach is less certain due to the uncertainties of the present knowledge of g ( r )at short distances, conclusions on the effective value of the mean curvature of theintermolecular potential in the liquid phase are essentially the same as those obtainedfrom Prof.Rowlinson’s argument.Prof. C. Domb (King’s College, London) said : Despite the considerable progressin our information regarding intermolecular potential energies, our knowledge of therepulsive part of this potential still remains inadequate. I should like to drawattention to a macroscopic property which should be capable of yielding informationon this aspect. I refer to melting data at high pressures ; although these have beendiscussed by Prof. Rowlinson quite recently,2 I think that more detailed investigationis desirable.Great experimental care is needed to establish pVT relations under pressure.It is much easier to decide whether a substance is in the solid or fluid state, and henceextensive data exist for a variety of substances over a considerable pressure range.Thus, for more than 10 years data for the melting curve of solid argon have beenavailable for pressures up to 10,000 atm.3 More recently, melting curves have beenobtained for a number of substances up to 100,000 atm.4 The new shock-wavetechniques developed in the University of California have opened up promise of dataup to 500,000 atm.5 At this last pressure solid argon is compressed to some 40 %of its equilibrium volume.It is clear that the data contain information about re-pulsive forces in the solid state, and our problem is to find a suitable method of extract-ing this information.I should like to suggest using the Lindemann melting formula,T,/MVi02 = constant.(1)The formula works satisfactorily for a wide variety of substances.6 Salter has shownthat the Simon melting formula, which works well for melting curves under pressure,can be derived with reasonable assumptions from the Lindemann formula.7 Atheoretical basis has also been provided for the formula.8Suppose we now have a measured melting curveP m = f(Tm)-We can use a Debye approximation for the solid to obtainprn= -dU,/dV-F(O, dO/dV), (3)where UO is the static lattice energy and F i s a known function.It is reasonable totake only nearest neighbour interactions into account at high pressures and then 0can be simply expressed in terms of d2UOldV2. Equating the values of pm in (2) and(3), and eliminating Tm by (1) we obtain a differential equation for UO as a functionof V. The equation is complicated and non-linear, but modern computers shouldenable the integration to be performed without too much difficulty.It would also be desirable to test the formula (I) directly under pressure, and thiscan most readily be achieved by measuring the velocity of propagation of longitudinal1 Physica, 1964, 30, 937.2 Rowlinson, Mol. Physics, 1964, 7, 349 ; 8, 107.3 Robinson, Proc. Roy. SOC. A , 1954,225, 393.4 Strong, Progress in very High Pressure Research, (John Wiley, 1961), p.182.5 Alder and van Thiel, Physics Letters, 1963, 7, 317.6 Gilvarry, Physic. Rev., 1956, 102, 308, 317.7 Salter, Phil. Mag. 1954, 45, 369. * Domb, Nuovo Cimento, suppl., 1958, 9, 1GENERAL DISCUSSION 55and transverse waves in a solid under pressure near the melting point. The onlydirect test of the formula (1) under pressure so far is for solid helium by Dugdale andSimon.1 The results were encouraging; but helium is anomalous because of thelarge zero-point energy and corresponding tests are needed for normal substances.Prof. R. B. Bernstein (University of Wisconsin) said : It is gratifying to see that thelong-range 0 6 ) coefficients derived from low-temperature transport data by theseextrapolation techniques are in good accord with the theoretical values, i.e., thatthe long-standing discrepancy of a factor of two (arising from the use of the " best fit "L.-J.(12,6) parameters : 4&&= 0 6 ) ) has now disappeared. In the same connectiona similar discrepancy in 0 6 ) values derived from molecular-beam scattering measure-ments at thermal energies has also been cleared up, and such data can now be con-sidered to yield reliable information on the long-range part of the interaction potential.The most recent beam scattering measurements 2 of absolute total cross-sections(ca. 5 % accuracy) for elastic scattering of Ar-Ar, Ar-Kr and Kr-Kr yield C(6)constants which agree with the theoretical values 3 to within an uncertainty (about15 %) consistent with the above-mentioned inaccuracy in the cross-sections.Similar(but somewhat less reliable) results for several alkali-metal rare gas systems havealso been reported.4Also, although absolute values of thermal molecular beam scattering cross-sections(and the C(6) values derived therefrom) have been until recently quite uncertain,d a t i v e values have been fairly precisely known and are well correlated with theory.For example, in a study 5 of the scattering of atomic beams of potassium by someseventy-eight molecules of varied complexity (and of caesium by sixteen differentmolecules) it was found that the 0 6 ) constants calculated by the Slater-Kirkwoodequation (with an added Debye induction term when appropriate) reproduced theexperimentally derived relative C(6) values with an average deviation of about 20 %.If one excludes from consideration the lighter scattering gases (for which the inter-pretation of the cross-sections in terms of C(6) is not correct, due mainly to the in-fluence of the short-range part of the potential) the average discrepancy betweenexperimental and theoretical 0 6 ) values (for 82 different interacting pairs) is reducedto 10 % (a figure well within the expected range of reliability of the Slater-Kirkwoodformula).Dr.D. A. Jonah and Prof. J. S. Rowlinson (Imperial College, London) said: Dr,Varshni has shown how difficult it is to fit the second virial coefficient of helium to atheoretically determined potential. We have now carried out our suggestion ofobtaining the repulsive part of this potential directly from the results of Yntema andSchneider without the prior assumption of any model, theoretical or empirical.We first applied a small correction to their results from 273" to 1473°K to allowfor the effect of the small negative part of the potential and fitted the correctedmeasurements towhere A = 24.44 cm3 deg.1'10 mole-1, v = 1/10, y = 2.257 x 10-4 deg-1.Theinverse Laplace transform of (1) gives 1-3 as a function of u, viz.,B = AT-" exp (- y T ) , (1)3nNr3 = A(ky/u)*'J?,(2y%.8),where J-,, is a Bessel function of negative order.1 Dugdale and Simon, Proc. Roy. Soc. A, 1953,218,291.2 Rothe and Neynaber, J. Chern. Physics, in press.3 Kingston, Physic. Rev. A, 1964, 135, 1018 ; Barker and Leonard, Physics Letters, 1964, 13, 127.4 Rothe and Neynaber, J.Chern. Physics, 1965, 42, 3306 ; Florin and Pauly, unpublished.5 Rothe and Bernstein, J. Chern. Physics, 1959, 31, 161956 GENERAL DISCUSSIONThe potential (u/k) of (2) is similar in the range 300-1500°K to that obtained byYntema and Schneider by conventional methods. It is a little less steep above 1000”K,a difference which leads to a better fit of the experimental results. However, themain purpose of this communication is to show that, at least for helium, the inverseroute is practicable and avoids the necessity of choosing a model.Dr. A. E. de Vries (Amsterdam) said: It is usually assumed that even for diatomicmolecules the Chapman-Enskog theory is reasonably accurate. We found, however,from thermal diffusion experiments with different isotopic CO molecules that onehas to be very careful. The inversion temperature, which for a given model shouldbe directly related to the size of the potential well, was found to be very differentfor the three mixtures 12C180/12C160, 14C160/12C16O and 14C160/12C180.Wefound 110, 175 and 250”K, respectively.1The dependence of the diffusion constant on temperature, calculated by means offormula (4) gave the same value for all pairs, within 1-2 % and was in agreement withexperiments by Amdur and Shuler.2It appears that this behaviour is not caused by the occurrence of inelastic collisions ;the pairs 14N15hT/14N2 and 15N2/14N2 have the same inversion temperature, althoughthe difference in inelastic collision behaviour in this case is more pronounced than inthe CO case.3 It also was proved theoretically that the influence of inelastic collisionson thermal diffusion is very small.4Dr. R.Duren and Dr. Ch. Schlier (University of Freiburg, Germany) said : For highenergies the deflection angle can be written (as Beck has shown) aswhere K = E/&V = dV/dr and b is the impact parameter. For heuristic purposesone can approximate further and writewhich seems to be the simplest non-trivial approximation possible. Our group inFreiburg is doing some tests to find out the usefulness of the latter approximation.RAINBOW SCATTERING.-The deflection function (1) for the rainbow angle meansthatwhere ri denotes the position of the inflection point of the potential.One knowsthat OR- 1/K if K>5. The new formula says that - @ ~ K ~ r i V ’ ( r i ) . As fig. 1 showsthis is not well fulfilled. However, a plot of - BRK against riV’(rt) yields one, almoststraight, line for different potentials.This means that whereas one cannot directly deduce any of the usual potentialparameters by the measurement of the rainbow angle only, one cun deduce with goodaccuracy the value of riV’(ri) or (since re is nearly a constant) V’(ri).B(b) M - bV’(b)/K, (1)i3,K = - riVt(ri),AMPLITUDE OF UNDULATIONS IN IMPACT SPECTRA.-starting from the expressionwhere (Ie is the average cross-section in high-energy approximation, one gets by1 de Vries and Boersma-Klein, Physica, to be published.2 Amdur and Shuler, J. Chem. Physics, 1963,38,188.3 de Vries and Haring, 2.Naturforsch., 1965, 20% 433.4 Baranowski and van de Ree, Physica, to be publishedGENERAL DISCUSSION 57inserting eqn. (1) and the conditions of the high-energy limit,A€?* l-lJv"(r,>,where AQ* = (Q(N = 1)- Q(N = 1-5))/Qs.With this relation (and without taking it literally) we hope to obtain a workingformula which could simplify the fitting of experiments. Trial computations havebeen started with the potentialV(X) = &(AX-" + Bx-" + C X - ~ ) with x = r/rm,where the quantities E , r, have their usual meaning. The coefficients A and B weredetermined in such a way that the usual conditions Y(1) = --E and V'(1) = 0 are met.I -7I+- 1---- ~ - -r--- - J 3 4 5ri Vi'0 2FIG. 1.-Dependence of -0RK upon riY'(ri) for different potentials and K p 5 .I I I I ' , ! , I10 20 30 40 50 60m+nFIG.2.-Dependence of AQ* upon m+n, i.e., the curvature of the potential well in the minimum.The solid line is --(m+n)*58 GENERAL DISCUSSIONWe present here only results with the restriction C = - 1 with the consequencesthat (i) A>@ B<O, (ii) V"(l)-nz+n. With the sun1 of n?+n constant one getsquite different potentials with equal curvature in the minimum, while varying thesum m+n should yield the relation in question. Fig. 2 shows the results obtained.AQ* is plotted against the sum m+n, i.e., the curvature of the potential well in theminimum. The solid line represents the 1/ ,/ V"(r,) law. It is seen that the curvatureis correlated within about 5 % with the undulation amplitude and the inverse squareroot law is borne out reasonably well.Prof.R. B. Bernstein (Ulziversity of Wisconsin) (cornnzzmicated): In 1942 Hirschfelderet al.1 pointed out the role of double molecules in the interpretation of gas imperfectiondata (particularly second virial coefficients), and suggested the use of spectroscopictechniques for their observation. Later, Bernardes and Primakoff 2 estimated (onthe basis of the best available L.-J. (12,6) potential parameters) the number N ofbound states of the rare gas dimers and predicted significant equilibrium concentrationsof dimers 3 (e.g., 2-3 % for Ne2 through Xe2 at T/TC = 2 and p/pc = 0.1) ; they sug-gested dimer detection via the Raman effect.It seems possible that the heteronuclear rare gas diatoms (with small but non-zerodipole moments) should be detectable by modern sensitive microwave or far infra-redabsorption techniques.Unfortunately, the uncertainty in the presently availablerare gas potentials (and the inaccuracy of the " combining rules ") is so great thatbound state predictions and estimates of spectroscopic constants are quite unreliable.However, this is no longer the case for the He-rare gas heteronuclear diatomicmolecules. Due to the availability of recent molecular beam scattering data for thesesystems,4 it now appears possible to make semi-quantitative estimates of some of thespectroscopic properties of these diatoms.From the velocity dependence of the total elastic scattering cross-sections (theextrema-effect 9, Duren et aZ.4 deduced the L.-J. (12,6) potential parameters ~ 1 2 , ~ ~ .(The constraint of the assumed L.-J. functionality imposes, as usual, a correspondingconstraint on the fitted parameters; however, this is thought to be of secondaryimportance.) From their potential constants the corresponding spectroscopicquantities (consistent with the L.-J.(1 2,6) function) are readily calculated ; the resultsare tabulated below.diatom 1 0 1 s x s l Z 9 re, A 4 D,, em-1 Be, cm-1 oe;cm-l N 6 608, cm-1 ; "K ex') erg 4HeNe 2.32 3.05 11.7 0.543 30.2 1 1.9; 2.8HeAr 3.50 3.45 17.6 0.389 31.4 1 5.9; 8.5He= 3-65 3-75 18.4 0-314 28.8 1 7.2 ; 10.4HeXe 3.85 4.05 19.4 0.265 27.2 2 8 . 5 ; 12.3Due to their greater stability (and smaller Be) the other heteronuclear rare gasdiatoms (i.e., those not involving He) should be more readily detected and character-ized spectroscopically.71 Hirschfelder, McClure and Weeks, J. Chem. Physics, 1942, 10, 201. Curtiss and Hirschfelder,2 Bernardes and Primakoff, J. Chem. Physics, 1959, 30, 691.3Recently three-body bound states in the rare gases have been calculated to be important:Zickendraht and Stenschke, Physics Letters, 1965, 17, 243.4 Duren, Feltgen, Gaide, Helbing and Pauly, Physics Letters, 1965, 18, 282. From their resultsfor ~ 1 2 and r , the usual " combination rules " fare badly onq2 but are moderately satisfactory for rm.5 see, e.g., Bernstein and O'Brien, this Discussion.6 obtained with the use of the WKBJ integral tables of Harrison and Bernstein, J. Chem. Physics,7 see recent related work of Kudian, Welsh and Watanabe, J. Chem. Physics, 1965, 43, 3397.ibid., p. 491.1963, 38, 2135

ISSN:0366-9033    

DOI:10.1039/DF9654000053

出版商:RSC    

年代:1965

数据来源: RSC

摘要:

Effective Pair Interaction Energy in Crystalline ArgonBY M. L. MCGLASHANDept. of Chemistry, The University, ExeterReceived 30th June, 1965Following the theoretical analysis of Guggenheim and McGlashan (1960) the determinationof the shape of the effective pair interaction energy in the neighbourhood of its minimum for solidargon is refined in the light of new experimental results and with the help of improved computingfacilities. The theory is shown to give a good account of all the relevant and reliable experimentalmeasurements on the solid. It is shown that there is no need to seek any special explanation ofthe rapid rise in the measured heat capacity of solid argon at temperatures just below the triplepoint.PREVIOUS WORK AND PRESENT PROGRAMMEThe dependence of the interaction energy w of a pair of Group 0 atoms on theinteratomic distance r can be expressed in the neighbourhood of its minimum atro as a power series in (r/ro- 1)¶ namely,w = - E + ~ ( r / r o - 1)2- a(r/ro - 1)3 +/3(r/ro - 1)4 - .. . (1)In the crystal the distance between nearest neighbours is approximately 10, and thatbetween next-nearest neighbours is approximately 1 -410. For next-nearest neighboursand a fortiori for more distant neighbours w is given almost certainly with sufficientaccuracy bywith X determined by quantum-mechanical calculations.In 1960, Guggenheim and I published 1 the results of an investigation into howfar the shape of the pair interaction energy, that is to say, the parameters of eqn. (l),could be determined for argon from the available experimental results for the equi-librium properties of the crystal.An outline of our method follows; for fullerdetails the reader is referred to the original paper.w = -X(ro/r)6 (r>l*4ro) (2)The energy of the crystal per atom when every atom is on its lattice site is~ { - E + K A ~ - u A ~ + / ~ A ~ - . . . -J&l+A)-6}, (3)where A = (R/ro- 1) and R is the distance between nearest neighbours in the crystal,and where Jn = (Cn/12- 1) and C n is the crystal potential constant 2 for power-n in a face-centred cubic lattice.According to first-order perturbation theory 3 each vibrational energy level ofa three-dimensional oscillator is the sum of three terms, corresponding to motionin three mutually perpendicular directions, and each having the form(n + +)in, + (n2 + n f +)ykT, (4)where v is the characteristic frequency of the oscillator and y is the anharmonicfactor. For any atom we evaluate v and y by calculating the change of energywhen the atom undergoes a small displacement from its lattice site, all the otheratoms being regarded as at rest on their lattice sites.Our theory thus neglects560 PAIR INTERACTION ENERGY I N CRYSTALLINE ARGONcorrelations among the motions of the atoms and implies only a single (Einstein)frequency. The results of these calculations 1 arev 2 = (1 2/2.x2m rg){ K( 1 + A)- (1 + 3 4113 - a A( 1 + A ) - '(I + 2 0 ) +y = -(3h2/8n4m2r~v2kT)(a(l+d)-' -B(l +A)-'(1+5d)- . . .where m is the mass of an atom.+2~A2(1+A)-'(l+5A/3)- .. . -5J,X(l+A)-'), ( 5 )+70Jl0x(1 +A)-''), (6)The corresponding vibrational partition function Q is given byIn Q = - 3 In (2 sinh +x) - 3y 3 coth2 &x,where x = hv/kT and where we have (but compare ref. (4)) omitted spurious termsin y2.(7)The molar Helmholtz function F of the crystal is then given byF/RT = (12/2kT)(-~+ud2--ad3+~A4- . . . -J&l +A)-6>+ 3 In (2 sinh +x)+3y + coth2 +x. (8)Formulae for all the equilibrium properties of the crystal can be derived from eqn.(8) which is for given values of the parameters E , K, a, p, ro, and h an explicit functionof T and R, or of T and V since the molar volume V is related to R by the formulaV = L J2R3, where L is Avogadro's constant. Experimentalists, however, usuallychoose T and p as independent variables.In calculations for comparison withexperiment we must therefore first evaluate R (or V ) at given T andp by use of theequation 1 for p obtained by differentiation of eqn. (8) with respect to Vat constant T.Eqn. (8), and formulae derived from it after differentiations with respect to Tor V, will be referred to as the anharmonic approximation when y is given by (6),and as the harmonic approximation when y is put equal to zero.The present programme differs in several ways from that carried out in ourprevious work.1(a) We make use of new experimental results which are presumably much moreaccurate than those which were available to us in 1960.(b) As previously, we neglect all the terms beyond PA4 in eqn.(1). We havetherefore to discuss values of the parameters E , K, a, 8, X and ro. In contrast to thesituation in 1960 when we were forced to assume a value for /3 (we actually assumed#I = 0 or alternatively /l = a), the new experimental results and our improved com-puter facilities now make it possible to find out more about the value of p.(c) Most of our previous work was based on the harmonic approximation.We showed, however, that the differences between values calculated according tothe harmonic and the anharmonic approximations are small when /3 is approximatelyequal to a, but not otherwise. Having at that time no access to information aboutthe value of /I we therefore put equal to a, which is equivalent to assuming thatthe effects of anharmonicity are unimportant, and showed that values of K, a, roand E could be chosen so as to give a fairly good fit of the then available experimentalresults when either the harmonic or the anharmonic approximation was used.Here we shall base our analysis on the anharmonic approximation making use ofthe harmonic approximation only to check that the difference between the twois not so large as to destroy confidence in the results of first-order perturbationtheory.( d ) Instead of our previous rough fit, made laboriously on a desk calculatingmachine, we now choose the parameters K, a, /3 and ro by optimizing the fit ofexperimental values by eqn. (8) and equations derived from itM.L . MCGLASHAN 61(e) Instead of the several values of A/k ranging from 100 to 200X which we triedpreviously, we now adopt the value A/k = 143°K derived on the assumption thatYO r 3.815 A from a new quantum-theoretical value h g / k = (4-41 +O-1) x 105 A6 OKcalculated by Dalgarno and his colleagues.~(f) Having found best values of K , a, p and YO, we shall use measurements of thevapour pressure of the solid, as well as the calorimetrically determined enthalpyof sublimation, to find the value of E .(g) In our previous work we integrated the experimental values of the heatcapacity at constant pressure and then compared the resulting entropies and en-thalpies with the theory.Here, however, we shall compare the experimental heatcapacity at constant (effectively zero) pressure directly with the theory. We dothis for two reasons.First we wish to draw attention to our view that it is theactual measured quantity which should be directly compared with any theory, andespecially never any quantity, such as the heat capacity at constant volume orGriineisen’s constant, obtained from the actual experimental results by the notoriouslyinaccurate process of differentiation, sometimes even carried out more than onceand usually made worse by dubious smoothing. Secondly, we wish directly to refutean illogical inference which has been drawn about the dependence of the heat capacityat constant pressure on temperature at temperatures just below the triple-point.EXPERIMENTAL DATA A N D PROCEDUREThe relevant measurements available on argon are : 196-9(1) calorimetric measurements of the heat capacity of the solid at effectively(2) measurements made by X-ray diffraction of the interatomic distance in the(3) measurements of the molar volume of the solid made directly at effectively(4) piezometric measurements of the decrease of the molar volume of the solid(5) measurements of the velocities of longitudinal and of transverse waves in(6) measurements of the equilibrium vapour pressure of the solid, and(7) calorimetric measurements of the enthalpy of fusion at the triple point,of the heat capacity of the liquid, and of the enthalpy of vaporization at the normalboiling point, and measurements of the second virial coefficient of the gas, whichlead to a value for :(8) the enthalpy of sublimation of the solid at its triple point to atoms at restat infinite dispersion.We use only (1) and (2) to evaluate best values of the parameters K, a, /3 and YO.(These quantities like (3), (4) and ( 5 ) depend on K, a, /3, X and YO but not on 6.) Weare insufficiently confident of (3), (4) and (5) to make use of them for this purposebut, having fixed K, a, and YO by optimizing the fit of (1) and (2), we then comparethe experimental values of (3), (4) and ( 5 ) with values calculated according to thetheory.We do not use (6) in the determination of the parameters K, a, /3 and robecause this quantity involves also the parameter E which we prefer to considerseparately. The quantity (8) also involves the parameter E . Having fixed the para-meters K , a, /3 and ro we fix E by comparison both with (6) and with (8).Finally,we use the values of K , a, j3, ro and E to compare the temperature dependence of(6) with that predicted by the theory.zero pressure ;solid at effectively zero pressure ;zero pressure ;on raising the pressure ;the solid at effectively zero pressure 62 PAIR INTERACTION ENERGY I N CRYSTALLINE ARGONOur computer programme can be used for either the harmonic or the anharmonicapproximation. It selects the values of K, a and ro which give the best overall fitof any number of measurements of the heat capacity and of the interatomic distancefor a specified value of /3 and a specified value of A. The formula for the heat capacityCp at effectively zero pressure is derived from eqn. (8) for the Helmholtz function byuse of the relationC, = - T(a2F]8T2)y 4- T(a(aFJaV),/aT)~/(a2F/aV2)~ (9)with the value of the molar volume V at effectively zero pressure determined bysolutions of the equationThe formula for the interatomic distance R at effectively zero pressure is just eqn.(10) together with the relation Y = LJ2R3.As already explained the value of h was fixed at 143 k"K.The value of fi wasvaried in the hope that (i) a value (or range of values) would be found which givesa significantly better fit than any other, and (ii) that this value would not implytoo large a difference between the harmonic and anharmonic approximations, thatis to say, that p/a would be not far from unity.The measurements of Cp and of R actually used in the fitting procedure werefor Cp those of Flubacher, Leadbetter and Morrison 10a at temperatures between10°K and the triple-point temperature 83.81"K, and for R those of Peterson,Batchelder and Simmons 11 between 60 and 83°K and those of Barrett and Meyer 12between 15 and 64°K.We regard these measurements as having superseded allprevious measurements of Cp and of R. Measurements at lower temperatureswhether by these or other authors were not used in the fitting procedure because itis certain 1 that at sufficiently low temperatures the Einstein approximation mustbreak down. For this reason we were tempted to assign to the data weights increas-ing with the temperature but decided not to do so. A trial calculation in which thedata were assigned weights proportional to the temperature was, however, carriedout.It led to values of the parameters, and to an overall fit, gratifyingly littledifferent from those obtained with the unweighted data.The parameters K, a, 18, ro and h having been fixed, we fix E by each of twoindependent methods as follows.(a) The equilibrium vapour pressure ps of the solid at temperature T is relatedto the Helmholtz function P by the formula 13(11)where F has the same zero (atoms at rest at infinite dispersion) as in eqn. (8), M isthe molar mass (39.948 g mol-I), B is the second virial coefficient, and Y is the molarvolume of the solid.(12)(dF/dV), = 0. (10)la (p'latm) = F/RT+$ In (T/4-3318"K)+$ In (M/g mol-')-p'(B- V)/RTFor B we use the formula 14B/Vc = 0.440f 1.40(1 -exp (0-75Tc/T)),where Yc is the critical volume (75.2 cm3 mol-1),15 and Tc is the critical temperature(150.7"K).ls An experimental value of p s at temperature T is used to calculate anexperimental value of the unknown quantity F/RT in eqn.(11). We next useeqn. (8) with our values of K, a, p, YO and A, and with R (or V ) determined by solutionsof the equationwhich at the relevant pressures differs insignificantly from eqn. (lo), to calculatep s = - (aF/aviT (13M. L. MCGLASHAN 63a theoretical value of (E/RT+~E/~T). The value of E is then obtained by simplearithmetic.(b) An experimental value of the energy U of the solid at the triple point (Tt, pt),referred to a zero of the atoms at rest at infinite dispersion, is obtained by use of theformula *u = - ptV: - AfH( K) - f b CbdT - AeH( Tb) + $RTb + pb(B - TdB/d T)T = T b , (14)Ttwhere V; is the volume of the solid at the triple point, AfH is the enthalpy of fusionat the triple point, CL is the heat capacity of the liquid, AeH is the enthalpy of vapor-ization at the temperature Tb, andpb is the vapour pressure of the liquid at Tb.Wetake pt = 0.680 atm,lOa V; = 24.65 cm3 mol-1,11 AfH/R = 143~2°K (we regardthisvalueloaas supersedingprevious measurements), Tb = 87.30"Kwhen P b = 1 atm, lon Jll (C',/R)dT = 17.4"K(for this least certain of the terms in eqn. (14) we adopt the older measurements ofClusius 16 rather than the more recent measurements of Flubacher et aZ.loa),AeH/R = 783.8"K,1Oa3 17 and B as given by eqn. (12), and obtain the value U/R =-735~3°K.We next use the formulaU = - T2(a(F/T)/aT}, (15)with P given by eqn. (8), with our values of K, a, 18, YO and A, and with R (or V ) Geter-mined as described above, to calculate a theoretical value of (U/RT+6c/kT) at Tt.The value of E is then obtained by simple arithmetic.The parameters K , a, /?, YO, h and E all having been fixed we finally calculatetheoretical values for the equilibrium properties and compare them with experi-ment. In particular we calculate Cp from eqn. (9) and (lo), R(p = 0) or V(p = 0)from eqn. (lo), R(p>O) or VCp>O) from the formulathe quantity (u:-$u;)s, where ul is the velocity of longitudinal waves and L+ is thevelocity of transverse waves, from the formulap = -(aF/av)T, (16)v; - f f = - ( V2lM>@PP v.9= (v2/M)[(a2F/av2)),- { a ( a ~ l a v ) ~ / a ~ ) F / ( a ~ ~ j a ~ ~ ) ~ ] (17)and the vapour pressure ps of the solid from eqn.(1 1 ) with F given by eqn. (8) and(13).RESULTS AND DISCUSSIONThe best fit of the experimental results is obtained with values of /?/a between1.25 and 1.50. When /3/a is given values progressively less than 1.25 the fit steadilyworsens but is still fair at /3/a = 1-00 in agreement with our previous work.1 Whenp/a is given values greater than 1-50 the fit rapidly worsens and is already bad for,8/a = 1-75. No doubt the best value of #3 could be further localized by calcula-tions for values between 1.25~ and 1.50~. In the meantime we give in the table thevalues of E , K, a, 18, X and YO for = 1.25~~ and for /3 = 1.50a.The values obtainedfor ~ / k are for /3 = 1-25a by method (a) 140-0°K and by method (6) 140.2"K, andfor p = 1.50~ by method (a) 140.7"K and by method (b) 140.8"K. The agreement* Formula (4) given in ref. (106) and copied by Pollack 8 is incorrect64 P A I R INTERACTION ENERGY I N CRYSTALLINE ARGONis well within the experimental uncertainties. In all the diagrams which we shallnow describe, the theoretical curves are those calculated with = 1.25a, that is tosay, with the parameters in the first row of the table.Fig. 1 shows the calculated and the observed values of the heat capacity of solidargon at effectively zero pressure. In order to avoid overcrowding the diagramwe have plotted only the smoothed experimental results of Morrison and hisTABLE VA VALUES OF TEE PARAMETERSelkOK ic/k°K u/k°K Plk°K yk0K ro/A140-1 465 8 22090 27610 143 3-801140-8 4815 24700 37050 143 3.792colleagues 10a at round temperatures.These, however, faithfully represent theirsplendid measurements, The agreement with experiment is satisfactory andparticularly so at temperatures approaching the triple-point temperature where Cpincreases rapidly with increasing temperature. This rapid increase has been widelythought to require some special explanation such as “ premelting” and has beenwidely ascribed to the thermal formation of vacancies in the crystal lattice. loby 18, 8 9 976 - I“0 5- E*I 9 4-G 3 -.3 cd12 -1 --1 I‘0 10 20 30 sb ;(I i0 90T/”KFIG. 1 .-The heat capacity C, of solid argon at effectively zero pressure plotted against temperatureT.The curve is that calculated according to the theory as described in the text. The circles arethe smoothed experimental results of Flubacher, Leadbetter and Morrison.loaWe see that there is no need to seek any such explanation or to invent vacanciesin order to describe the equilibriuni properties of the solid. Moreover, Simmonsand his colleagues 11 have compared their X-ray interatomic distances with macro-scopically determined molar volumes and conclude that if there are any vacanciesat all then even at the triple point these are present only at far lower concentrationsthan were assumed by the authors of ref. (lob) and (18). They have also showM. L . MCGLASHAN 65that their X-ray results close to the triple point preclude " any significant pre-melting phenomena in pure argon ".Fig.2 shows the calculated and the observed values of the interatomic distancein solid argon at effectively zero pressure. The agreement of theory and experi-ment is again satisfactory. Even the discrepancies at temperatures below about40°K are in a sense satisfactory because we know that the Einstein approximationmust fail progressively at sufficiently low temperatures. In this connection werepeat that the data used in the fitting procedure were equally weighted.Fig. 3 shows the calculated and Stewart's 21 observed values of the dependenceof the molar volume on pressure at 65 and at 77°K. At both temperatures the dis-crepancies increase with increasing pressure.This must be expected. Eqn. (1)with the parameters given in the table is supposed accurate only over a range ofabout 0-25A on either side of its minimum at 3-80A. The combined effects ofpressure and of the amplitude of the vibrations of the atoms about their meanpositions on the crystal lattice ensure that at pressures greater than about 1000 atmwe are dealing with values of the interatomic distance corresponding to pointson the repulsive branch of the pair interaction energy which are outside that range.At pressures below 1000 atm the agreement is within the experimental accuracyclaimed by Stewart at 77"K, but not at 65°K. In view of the difficulty of experi-ments in which the volume change is measured when a polycrystalline mass of argonis compressed behind a piston we are more suspicious of the low pressure measure-ments at 65°K than of the theory.Fig. 4 shows the calculated and the observed values of the quantity ( v ~ - $ v ~ ) fat effectively zero pressure.Barker and Dobbs 22 measured vz at 67, 74 and 79"K,and ct at 67 and 74°K. They quote only smoothed and extrapolated values atround temperatures between 0 and 84°K. The three values shown as open circlesin fig. 4 are those calculated from their smoothed results at 60, 70 and 80°K. Jones66 PAIR INTERACTION ENERGY I N CRYSTALLINE ARGONand Sparkes 23 measured ut over the range 18-62°K. The value shown as a filledcircle in fig. 4 is that calculated from Barker and Dobbs’s v1 and Jones and Sparkes’sut at 60’K.Guptill, Hoyt and Robinson 24 measured uz at 64 and at 78°K. Thevalues shown as open squares in fig. 4 are those calculated from Guptill, Hoyt andI I% 77°K‘- ’ O L -19 I‘-0 2 4 6 8 10 12p/lOz atmFIG. 3.-The ratio (V@)- V(O))lV(O), where VCp) is the molar volume of solid argon at the pressurep, plotted against pressure at 65 and at 77°K. The curves are those caiculated according to thetheory as described in the text. 0, Stewart 2 1 ~ ; 0, Stewart. 21b*I I 1 10 20 30 40 50 60 90 80 900-8*T/”KFIG. 4.-The quantity (u;-$vi)*, where vi is the velocity of longitudinal waves and ut is the velocityof transverse waves in solid argon at effectively zero pressure, plotted against temperature T. Thecurve is that calculated according to the theory as described in the text.The experimental pointsare described in the text. The vertical lines correspond to experimental errors (k5 %) admittedby the authors.Robinson’s values of uz and Barker and Dobbs’s values of ut at 64 and 78°K. Thevalue shown as a filled square in fig. 4 is that calculated from Guptill, Hoyt andRobinson’s and Jones and Sparkes’s vt at 64°K. The agreement between theoryand experiment is probably as good as can be expectedM . L. MCGLASHAN 67Finally, fig. 5 shows the calculated and the observed values of the equilibriumvapour pressure of solid argon. The agreement of the theory with the results ofMorrison and his colleagues 10a is virtually perfect.1000=: 300200-x"2 2 'OO=Q#50-30-20[ I , - 1 I I 1 1 I 1 1 I I I I 1 - - - - - - \ - -"t., --- ---l o " ~ ~ " i I ~ i I I l l I I l 1 .12 13 14 ISCONCLUSIONSThe shape of the effective pair interaction energy in the immediate neighbourhoodof its minimum for solid argon is given with improved accuracy by eqn.(1) withparameters intermediate between the two sets given in the table. Our Einsteintheory with allowance for small anharmonic effects gives a good account of allthe relevant and reliable experimental measurements on the solid. There is noneed to seek any special explanation of the rapid rise in the measured heat capacityat temperatures just below the triple point.I am grateful to Dr. T. H. K. Barron, to Prof. A. Dalgarno, to Prof. G. 0. Jones,and to Prof.R. 0. Simmons for helpful exchanges and to Prof. E. A. Guggenheim forcharacteristically constructive criticism. It would not have been possible to completethis work in time for the Discussion without the help, far beyond the line of duty, of Dr.L. A. G. Dresel and Mr. C. D. Plews, of the University of Reading Computer Unit,to whom I am enormously grateful. In this connection I also warmly thank Mr.I. R. McKinnon of this University who checked all my algebra and spent manyfrustrating hours coping with defective computer programmes.1 Guggenheim and McGlashan, Proc. Roy. SOC. A , 1960,255,456.2 Lennard-Jones and Ingham, Proc. Roy. SOC. A, 1925, 107, 636. Hirschfelder, Curtiss and3 Pauling and Wilson, Introduction to Quantum Mechanics (McGraw-Hill, New York, 1933,Bird, Molecular Theory of Gases and Liquids (Wiley, New York, 1954), p.1040.formula (23.30), p. 16168 PAIR INTERACTION ENERGY I N CRYSTALLINE ARGON4 Henkel, J. Chem. Physics, 1955,23,681. Domb and Zucker, Nature, 1956,178,484. Zucker,5 personal communication from Prof. Dalgarno.6Dobbs and Jones, Reports Prog. Physics, 1957, 20, 516.7 Hollis Hallett, in Argon, HeZium and the Rare Cases, ed. Cook (Interscience, New York, 1961),8 Pollack, Rev. Mod. Physics, 1964, 36, 748.9 Boato, Cryogenics, 1964, 4, 65.10 (a) Flubacher, Leadbetter and Morrison, Proc. Physic. SOC., 1961, 78, 1449 ; (b) Beaumont,Chihara, and Morrison, Proc. Physic. Soc., 1961, 78, 1462.11 Petersen, Batchelder and Simmons, typescript, personally communicated, of a paper entitledObservations on the Thermal Defect Structure of Solid Argon which is to be submitted toPhil. Mag. This typescript contains a smoothing formula for R against T but no rawresults. However, a plot supports their claim that the formula fits their experiments withintheir extremely low experimental error.Phil. Mag., 1958, 3, 987.vol. 1, p. 313.12 Barrett and Meyer, J. Chem. Physics, 1964, 41, 1078.13 Guggenheim, Thermodynamics (North-Holland, Amsterdam, 1957), 3rd ed., formula (4.58.3).14 Guggenheim, Applications of Statistical Mechanics (O.U.P., Oxford), in press.15 Din, Thermodynamic Functions of Gases (Butterworths, London, 1956), vol. 2, p. 146.16 Clusius, 2. physik. Chem. B, 1936, 31, 459.17 Frank and Clusius, Z. physik. Chem. B, 1939, 42, 395.18 Foreman and Lidiard, Phil. Mag., 1963, 8,97.19 Figgins, Ph.D. Thesis (London, 1957) ; see also Dobbs, Figgins, Jones, Piercey and Riley,20 Clusius and Weigand, 2. physik. Chem. B, 1940, 46, 1 ; see also ref. (1).21 Stewart, (a) Physic. Rev., 1955, 97, 578 ; (b) J. Physic. Chem. Solids, 1956, 1, 146.22 Barker and Dobbs, Phil. Mag., 1955, 46, 1069.23 Jones and Sparkes, Phil. Mag., 1964, 10, 1053.24 Guptill, Hoyt and Robinson, Can. J. Physics, 1955, 33, 397.25 Clark, Din, Robb, Michels, Wassenaar and Zwietering, Physica, 1951, 17, 876.Nature, 1956, 178, 483

ISSN:0366-9033    

DOI:10.1039/DF9654000059

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年代:1965

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Interatomic Potentials in Ideal Anharmonic CrystalsBY T. H. K. BARRON"Dept. of Physical Chemistry, The University, BristolReceived 5th July, 1965A brief review is given of the information about interatomic forces at present obtainable fromthermodynamic and other vibrational properties of crystals, with particular reference to the inertgas solids. In these solids, anharmonic effects are appreciable, but for many properties they canbe taken into account by effective shifts in the normal mode frequencies. As yet, both theory andexperiment are insufficiently precise to indicate uniquely the influence of many-body forces.1. INTRODUCTIONIn this paper I discuss the kind of information about interatomic forces whichcan be obtained from the experimental properties of ideal crystals.Since thereare already many accounts of the harmonic theory and its applications,l-4 I shallconcentrate mainly upon the anharmonic theory, which has been systematicallydeveloped during the last decade by Leibfried and Ludwig59 6 Maradudin 7-10and others. In this context we require two things of an anharmonic theory: toenable us to correct for anharmonic effects when deriving information from essentiallyharmonic properties, and also to enable us to derive additional information fromthe anharmonic properties themselves.For an ideal crystal, the bulk properties depend only on the crystal potentialenergy function @[u(l), u(2), . . . u(M)], where the u(i) are small displacements ofthe atoms from their ideal lattice positions; this function depends on the macro-scopic strain of the crystal.It can be expanded as a Taylor series :@ = @(0)+@(2)+@(3)+@(4)+ * 3where @* is the static lattice energy andQ(3) = C Q2ia;jb;kyUa(i)UP0>Uy(k), etc* (1 -3)6ia: jb;kyThe coefficients @ i a ; j b , Q , ; j b ; k y , . . . are the crystal " force constants ", whichtogether with the atomic masses Mi determine the atomic vibrations and the thermo-dynamic properties, and play an important role in the interaction of the crystalwith radiation. Measurement of crystal properties can therefore give informationonly about @O and the crystal force constants, including their strain dependence.?Usually the amplitude of the atomic vibrations is sufficiently small for the suc-cessive terms of eqn.(1.1) to be of diminishing importance. Thus @(o) is a firstapproximation to the sublimation energy of the crystal, which is modified by takingvibrational energy into account (determined primarily by @(z), and to a lesser extentby higher terms). On the other hand, some crystal properties depend even in thefirst approximation upon terms higher than @(o); e.g., the spectrum of normal* Present address: Department of Theoretical Chemistry, The University, Bristol 2.7 The surfaces, vacancies and other imperfections of real crystals can give additional information.670 INTERATOMIC POTENTIALS IN CRYSTALSvibrations (and derived properties such as the entropy) depends primarily on Qi)(Z).Consideration of each new term modifies results derived from the previous terms,and also may give fresh properties.The harmonic theory neglects terms beyond CD(z).The atomic motions can thenbe resolved into independent harmonic waves whose frequencies and directionsof polarization depend only on the Mi and the @ga;jb. It is seldom if ever possibleto derive these constants accurately from fundamental quantum theory, but theycan be estimated from experimental data by fitting suitable models which limitthe number of independent force-constants. During the last ten years the use ofneutron 11 and diffuse X-ray 12 scattering to determine dispersion curves for wavespropagated along symmetry axes (together with high-speed digital computers forthe theoretical analysis) have given a great mass of detailed information about theinteratomic forces in many simple crystals.13 Force-constants estimated for alarge number of simple metallic, ionic and valence crystals show that appreciableinteraction usually extends to many neighbours of an atom and that non-centralforces are important.Much less is known experimentally about van der Waalscrystals, chiefly because of the difficulty of obtaining and handling single crystalslarge enough for neutron diffraction or elastic constant measurements.The heat capacity and other thermodynamic properties depend on averages overthe entire distribution of frequencies, and give less detailed information than theindividual frequencies derived from spectroscopic measurements. They are oftenexpressed in terms of equivalent Debye temperatures; e.g., W ( T ) is the character-istic temperature of the Debye distribution that would give the same value of CVat temperature T as the experimental solid.Modern calorimetry can give heatcapacities to within about 0.2 %, and agreement with experiment at all temperatures(and especially far T5W/3) is an exacting criterion for any force-constant model.We can also analyze the heat capacity 14 and the entropy,ls without reference toany specific model, to obtain directly other averages over the frequency distribution.For example, in this way we can calculate the zero-point vibrational energy, and soderive the static lattice energy @(o) from the sublimation energy at 0°K.2. QUASI-HARMONIC THEORYMany crystal properties which are dependent on CD(3) to the first order can tothis approximation be treated by the quasi-harmonic approximation, in whichanharmonicity enters only through the strain-dependence of @(2).* We can thusaccount for the pressure-dependence of lattice frequencies (e.g., as measured byinfra-red spectroscopy), for third-order elastic constants, for thermal expansionand for the temperature-dependence of second-order elastic constants.6The advantage of this theory is that although it deals with anharmonic pro-perties, it retains intact the framework of the harmonic approximation, and inparticular the simple concept of a harmonic vibrational spectrum.The strain-dependence of individual frequencies (depending ultimately on the strain dependenceof the igfa;,b) thus plays a vital role, and is conveniently described by dimensionlessparameters of a type first used by Gruneisen.The most widely used are those referringto a uniform expansion, defined by 16(2.1)where mj is the frequency of the jth normal mode. Direct measurement of the y jfor waves propagated along the crystal symmetry directions would determine the= 0, n>2), @(2) can be shown to be independent ofstrain.yi = -d In mj/d In V ,* When @ is purely harmonic (i.e.T. H . K . BARRON 71volume-derivatives of the @ i z ; j p , but this requires accurate neutron spectroscopyunder pressure. However, provided that we work with a model in which thereare only a small number of independent force-constants, we should in principle beable to deduce their volume-derivatives from much less complete information.For a small number of solids the pressure-derivatives of the elastic constants areavailable, but the commonest source of information about the yf is the thermalexpansion, given by 16here xiso is the isothermal compressibility and c3 the contribution of thejth modeto Cv.Although accurate measurements of p in the important low temperaturerange have been made for a large number of simple solids during the last few years,16comparatively little has yet been done to deduce the volume-derivatives of the @ i j ; j g .Without any assumptions about the number of independent force-constants,however, it is possible to analyze the experimental data so as to obtain informationabout the volume-dependence of the total frequency distribution, and hence alsoof crystal properties which depend on the frequency distribution.17 For example,we can estimate the volume derivatives of the zero-point energy Ez, and thus correctfor the dilation of the lattice at 0°K due to zero-point energy ; in this way we obtaina better estimate of the equilibrium spacing of the static lattice.The change in lattice frequencies due to thermal dilation can be deduced fromeqns.(2.1) and (2.2); to the first orderThis is a second-order effect in the y j and hence in @(3), and is of the same order ofmagnitude as other effects which cannot be treated by the quasi-harmonic theory(see following section). We cannot therefore determine the 73 simply from thetemperature-dependence of the q.3.GENERAL ANHARMONIC THEORYFor most solids the anharmonic terms @(3), 0(4), . . ., etc., are small, and canbe treated by perturbation theory. Because @(3) is an odd function of the displace-ments u,(i), its first-order contribution to most crystal properties vanishes ; theformal theory has therefore been developed systematically to the second-order termsin @(3) and the first-order terms in @(4), which give rise to effects of the same orderof magnitude. We shall consider effects at constant volume only, since we havealready seen that the change of lattice frequencies and related properties broughtabout by thermal expansion can be treated by the quasi-harmonic theory.It is not difficult qualitatively to predict the effect of an anharmonic perturbationon the classical motion of the harmonic crystal. The normal co-ordinates derivedfrom can still be used to describe the system, but their motion will no longerbe independent.Consider first a classical system. Because the interaction betweennormal co-ordinates is small, the motion of each will be approximately harmonic;a Fourier analysis of its motion will give a peak instead of the sharp line given bythe harmonic crystal, with a characteristic half-width r(q). We may also expectthe mean frequency of the peak to differ from the harmonic value, thus definingan anharmonic shift &oanh. Both width and shift will be temperature-dependent,because the magnitude of the interaction depends on the amplitude of the latticevibrations72 INTERATOMIC POTENTIALS I N CRYSTALSExperimentally, frequencies are determined by the energy absorbed or emittedby the crystal in " one phonon " processes, when it interacts with external radiation.In such experiments (e.g., with neutron or infra-red spectroscopy) temperature-dependent widths and mean frequencies are indeed observed.The detailed theoryof such processes 7, 18 gives the same formal expressions for 8cr)anh and T(o) whateverthe type of radiation used, and it is therefore straightforward to relate the resultsof different scattering experiments. Nevertheless, little information can be de-duced about the anharmonic force-constants because numerical calculations haveas yet been carried out only for the simplest models.On the other hand, the formal expressions can be used directly to correlate thefrequency shifts with anharmonic corrections to thermodynamic properties.Al-though a harmonic frequency distribution is totally inadequate to describe all thermo-dynamic properties, it is found that to a first approximation (second order in @(3),first order in @(4)) the entropy is given correctly when the mean (anharmonic) frequen-cies are substituted in the harmonic expression for the entropy 19 :S = Es(cdjh+ Gw;nh(T), T),jwhere mTh is the frequency given by the quasi-harmonic theory and s(cu,T) is the con-tribution to the entropy from a harmonic oscillator of frequency cr). From thisexpression the heat capacity can be obtained by differentiation, and the thermalcontribution to the Helmholtz energy by integration ; only the zero-point energyremains uncorrelated. The shifts 6cr)yh thus permit us to discuss even the anhar-monic contributions to the heat capacity in terms of a vibrational spectrum, and torelate this to spectroscopic properties.It can also be shown that in the limit of longwaves the anharmonic shifts determine correctly the anharmonic contribution to theisothermal elastic constants.The general expression for the shifts has the formwhere ezh is the quasi-harmonic contribution of mode k to E, ek,T and ek,Z its thermaland zero-point components, and A& and A& give the effects due to @(3) and @(4)respectively. At sufficiently low temperatures the terms in e k , z dominate, and thefrequencies are effectively independent of temperature ; in this range the heat capacitybehaves similarly to that of a purely harmonic crystal, although it depends on @(3)and @(4) as well as on (D(2).Thus even when the zero-point energy is relatively large(e.g., in the lighter rare gas solids) there is no obvious sign of anharmonic behaviourin the low-temperature heat capacity.20 At high temperatures, on the other hand,ezh = kT, giving frequency shifts and anharmonic contributions to S and Cv whichare directly proportional to the temperature :With sufficiently accurate data it is possible not only to detect anharmonic be-haviour, but even to estimate the value of A. A few such values are given in table 1,together with O z , the estimated quasi-harmonic limiting value of OC at high tem-peratures.It is found that A can be either positive or negative, and calculationson central force models 7,249 25 indicate that there is a balance between contribu-tions from @(3) and @(41, with @(3) tending to decrease frequencies and increaseCv, @(4) the reverseT. H. K . BARRON 73At lower temperatures it is not possible rigorously to separate anharmonic con-tributions from Cv without further information. However, the form of eqn. (3.2)suggests that as a first approximation we should assume the i l k to be all identical,so that A = -3NkA andi h Y h / ~ j = AEqh = - A(Eqh/3Nk). (3.4)Feldman, Horton and Lurie26 have shown how it is then possible to correct foranharmonic effects at low temperatures without making further assumptions.Crude though the approximation is, it is preferable to making no correction at allwhen estimating the harmonic force-constants from low-temperature data.Suchcorrections are needed whenever the zero-point energy is appreciable, and not onlyfor van der Waals crystals ; for example, they have been found necessary in account-ing for the vibrational properties of ice?‘r ABLE 1 -PARAMETERS FOR THE HEAT CAPACITY OF SOME SIMPLE SOLIDS AT HIGH TEMPERATURESAr 21 Pb 23 A 2 3 KCI 15; 23 NaI 1s@%(OK) 87 95 f5 420fl0 235fl 19531AOC, x 102 -4.5f2 -1.2f0.2 -1.lf0.2 0.7f0.2 -0.04f0.1Nevertheless, it is in departures from eqn. (3.4) that we can hope ultimately toobtain detailed information from the frequency shifts about the anharmonic force-constants.That significant departures occur is clear from Leadbetter’s discussionof the vibrational properties of lead.23 The negative value of A (table 1) indicatesthat the mean frequency shift (at constant volume) is positive; whereas analysisof the Debye-Waller effect would suggest that the constant volume shifts are negative.Since the Debye-Waller effect is dominated by modes at fairly low frequencies,this suggests that a substantial number of low frequency modes have negative fre-quency shifts. Preliminary neutron diffraction data 28 at constant pressure agreequalitatively with this result.Finally, in principle the variation of atomic mass (by employing different iso-topes) provides a powerful method of obtaining information about anharmonicforces.For example, the mass-dependence of the contributions of @(3) and @(4)to A is not the same, and it is found that at constant volume only @(4, contributesto the vapour pressure ratio of two isotopes at moderately high temperatures.29Unfortunately, such experiments have been sufficiently precise only to demonstratethe existence of such effects, and not to give quantitative information.304. RARE GAS SOLIDSQuantitative calculations of anharmonic effects have for the most part beencarried out for models with short-range central forces. Although they may throwlight on the behaviour of metals and other solids, they are most directly applicableto the rare gas solids. Even here, the existence of non-central forces cannot beneglected, and indeed one reason for accounting adequately for anharmonic effectsis to make it easier to identify the effects of non-central forces.31 Nevertheless,all theoretical estimates 32 indicate that the two-body central forces are much stronger,and it is probable that to a first approximation they determine the anharmoniceffects.Calculations of crystal properties at low temperatures 25 indicate that even fora Lennard-Jones 6: 12 potential anharmonic effects are dominated by the nearestneighbour interaction ; and for rare gas atoms the interaction between distan74 INTERATOMIC POTENTIALS I N CRYSTALSatoms is considerably less than is indicated by a conventional 6 : 12 interaction.33Anharmonic effects for the heavier rare gas solids should therefore depend primarilyon the shape of the pair potential 4(r) in the neighbourhood of its minimum #(ro),and in particular on the ratios r,+”’J4“ and rg4“”/4”.The 6 : 12 potential givesr0#“’/4” = -21 and ri#””/4” = 371, and similar values are found for a conven-tional 6-exp potential. More surprisingly, similar values (-21.8 and 391.2) arefound for the radically different potential of Munn and Smith,34 which, unlike the6 : 12 and the 6-exp, has the correct coefficient of r-6 at large distances. Anharmonicproperties calculated for any of these potentials should be approximately correctfor the others. If it is found that the calculations agree with available experimentaldata, we shall then have a model which can be used to correct for other anharmoniceffects which are not known experimentally-e.g., in the heat capacity at lowtemperatures.On the other hand, a serious discrepancy will suggest that the shapeof the pair-potential at ro is not given approximately by these potentials.Unfortunately neither theory nor experiment is at present sufficiently establishedto make a reliable comparison. There are three simple anharmonic propertiesavailable for comparison with theory-the Griineisen parameter, defined byy = p V / ~ a d C p , the coefficient A defined in eqn. (3.3), and the high temperaturethermal resistance. For the last of these there is as yet no reliable quantitativetheory. The Gruneisen parameter depends on the measured values of /3 and p d ,both of which may be affected by strains and imperfections in the solid, especiallyat low temperatures.The 6 : 12 potential predicts that at moderately high tem-peratures 7-3, which is in fair agreement with data on Ar, Kr * and Xe between40 and 60°K 32; but measurements on Ar and Xe at 20°K indicate a sharp drop iny which appears to be inexplicable on any central force model. In view of thesimplicity of the theory and the difficulty of the experiment, it is likely that this dropin y is spurious, and that 4“’ (ro) is fairly well represented by the 6:12 potential.Kuebler and Tosi 21 obtained A@: = - (4.5 +2-0) % from the entropy of argonby integrating C p and correcting to a constant volume by means of the relation( a s / a V ) ~ = p/xis*. Although the limits of error are wide, the theoretical value isat least equally uncertain.The best calculations as yet available of anharmoniccontributions to S are those of Maradudin et aZ.9. 10 but these neglect terms whichmore rigorous calculations at low temperatures25 show to be important. If,however, we make the assumption of eqn. (3.4), which is consistent with the termscalculated by Maradudin et aZ., we can obtain A from the low-temperature calcula-tions. For argon this gives A@%- - 11 % for a 6 : 12 potential, while the termscalculated by Maradudin et al. would alone give -22 %. If we take - 11 % asthe theoretical value, the discrepancy with experiment suggests that the value of+ I f f f predicted by the 6 : 12 potential is too large by about 20 %.It is clear that until more rigorous theoretical calculations are available at alltemperatures, we can hope neither to draw any firm conclusions about the mag-nitude of @ff‘ nor to be able to correct reliably for anharmonic effects at lower tem-peratures.It is therefore premature to attempt to identify the influence of many-body forces in these vibrational properties. Calculations at all temperatures havebeen recently carried out by Leech and Reissland, but from their preliminary report 35it is not possible to extract a value of A or fully to assess the approximations employed.I thank Dr. G. K. Horton, Dr. A. J. Leadbetter and Dr. R. J. Munn for discussingtheir results with me before publication.* The values for Kr in ref. (32) are based on incorrect values for the compressibility.2T .H . K . BARRON 751 Born and Huang, Dynamical Theory of Crystal Lattices (Oxford University Press, 1954).2 Blackman, Handb. Physik, 1955, 7, part 1, 325.3 de Launay, Solid State Physics, 1956, 2, 219.4 Cochran, Reports Prog. Physics, 1963,26, 1.5 Leibfried, Handb. Physik, 1955, 7, part 1, 104.6 Leibfried and Ludwig, Solid State Physics, 1961, 12, 275.7 Maradudin and Fein, Physic. Rev., 1962, 128, 2589.8 Maradudin and Flinn, Physic. Rev., 1963, 129, 2529.9 Maradudin, Flinn and Coldwell-Horsfall, Ann. Physics (New York), 1961, 15, 337, 360.10 Flinn and Maradudin, Ann. Physics (New York), 1963, 22, 223.11 Brockhouse and Stewart, Rev. Mod. Physics, 1958, 30, 236.12 Walker, Physic. Rev., 1956, 103, 547.13 Wallis (ed.), Lattice Dynamics, J. Physics Chem. Solids, 1965, suppl. 1.14 Barron, Berg and Morrison, Proc. Roy. SOC. A, 1957,242,478.15 Tosi and Fumi, Physic. Rev., 1963, 131, 1458.16 Collins and White, Prog. Low Temp. Physics, 1964,4,450.17 Barron. Leadbetter and Morrison, Proc. Roy. SOC. A, 1964, 279, 62.18 Cowley, J. Physics Chem. Solids, 1965, suppl. 1, 295.19 Barron, J. Physics Chem. Solids, 1965, suppl. 1, 247.20 Barron and Klein, Physic. Reu., 1962, 127, 1997.21 Kuebler and Tosi, Physic. Reu., 1965, 137, A1617.22 Flubacher, Leadbetter and Morrison, Phil. Mag., 1959, 4, 273.23 Leadbetter, in preparation.24 Maradudin, Fein and Vineyard, Phys. Stat. Sol., 1962, 2, 1479.25Barron and Klein, Proc. Physic. SOC., 1965, 85, 533.26Feldman, Horton and Lurie, J. Physics Chem. Solids, 1965, 26, 1507.27 Leadbetter, Proc. Roy. SOC. A , 1965, 287, 403.28 Brockhouse, in preparation.29 Boato, Casanova and Levi, J. Chem. Physics, 1962, 37,201.30 Klein, J. Chem. Physics, 1964, 41, 749.31 Wallace, Physic. Rev., 1964, 133, A153.32 Pollack, Rev. Mod. Physics, 1964, 36, 746.33 Munn, R. J., J. Chem. Physics, 1965, 42, 3032.34 Munn and Smith, J. Chem. Phy.uks, to appear.35 Leech and Reissland, Physics Letters, 1965, 14, 304

ISSN:0366-9033    

DOI:10.1039/DF9654000069

出版商:RSC    

年代:1965

数据来源: RSC

摘要:

Repulsive Energy in Sodium Chloride and Potassium ChlorideCrystalsBY E. A. GUGGENHEIM * AND M. L. MCGLASHAN 7Received 30th June, 1965The analysis of the repulsive energy in sodium chloride and potassium chloride crystals publishedfour years ago has been thoroughly revised, making full use of new and much more accurate ex-perimental data. The conclusions from the present analysis, in contrast to the earlier analysis.are entirely self-consistent and satisfactory.In 1961 a paper was published 1 having the same title as the present one. Thereader is referred to this paper for details of the notation used there and here.Meanwhile the situation has been completely changed especially by a recent ac-curate determination 2 of the electron affinity of chlorine and to a less extent by ac-curate measurements 3 of the thermal expansivity of potassium chloride over awide range of temperature.Consequently we are now able to make an accurateand reliable analysis of E and dE/d In R for both sodium chloride and potassiumchloride. We now dispense with any analysis of d2E/d (In R)2 since this requires theuse of old measurements of dK/dp in which we have no confidence.The new value of the electron affinity of chlorine gives at 298°KCl(g) + e-+Cl-(g), AH = - 84.82 kcal mole-1.This value includes a small term -0.02 kcal mole-1 to take account of C1 atoms inthe excited 2P+ state. We then have at 298"K,Na+(g) + CI-(g) +NaCl(s),K+(g) + C1-(g) -+KCl(s),AH = - 188-25 kcal mole-1 ;AH = - 171.42 kcal mole-1.We now give in tables 1 and 2 values for all quantities needing revision.TABLE ~.-REVISED ANALYSIS FOR NaClAIA-1 3.52 3.52TABLE 2.-REVISED ANALYSIS FOR KCIT/"K 298 2001 0 4 ~ 4 0 ~ - 1 1.12 1-00aTVK-l/kcal mole-1 5.22 3.32U/kcal mole-1 - 168.46 - 169.63- E,/kcal mole-1 172.09 172.14Er/kcal mole-1 19.88 20.61-dE,,/d In R)/kcal mole-1 16.5 10.9T/"K 298 200 100U/kcal mole-1 - 185.29 - 186.43 - 187.45- E&cal mole-1 189001 189-04 189.03EJkcal mole-1 22-61 23-45 24.153.52(dE,ld In R)/kcal mole-1 - 224.1 -231.5 -238.1000.761.3270.7 172.1421-365.8(dE,ld In R)/kcal mole-1 - 214.9 - 222.0 - 228.7(A/A--l 3.43 3-43 3-41* The University, Reading, England.t The University, Exeter, England,7E . A . GUGGENHEIM AND M. L.MCGLASHAN 77We conclude that for both salts Er can be accurately represented by the empiricalrelation Er = B exp (-AR) but that the values of A for the two salts differ by nearly3 % in contrast to the common assumption that they are equal.TABLE 3.-REVISED TEST OF GRUNEISEN'S APPROXIMATION FOR KClT/"K 298 200 100(aTVrc-l)/cal OK-1 mole-1 0.57 0.34 0.10C&al OK-1 mole-1 12-3 1 11.50 9.30C,/cal OK-1 mole-1 11.74 11.16 9-20y-1 = Kc"/av 0.673 0.668 0.693The accuracy of the new measurements of a for potassium chloride warrants anew test of Griineisen's approximation. This is given in table 3. We see thatthe quantity ~cC,/a V, which according to Gruneisen's approximation should be aconstant, in fact varies by 3 % over the temperature range 100 to 300°K. Recentexperimental measurements 4 show beyond doubt that y = a V/rcC, is far from beingindependent of temperature.1 Guggenheim, McGlashan and Prue, MoZ. Physics, 1961, 4,433.2 Berry and Reimann, J. Chern. Physics, 1963,38, 1540.3 Rubin, Johnston and Altman, J. Physic. Chern., 1962, 66, 948.4 White, Proc. Roy. Soe. A, 1965, 286, 204, fig. 2

ISSN:0366-9033    

DOI:10.1039/DF9654000076

出版商:RSC    

年代:1965

数据来源: RSC