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 . 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