Thermodynamics and Lattice Vibrations of Minerals, 3

VOL. 17, NO. I REVIEWS OF GEOPHYSICS AND SPACE PHYSICS FEBRUARY 1979

Thermodynamicsand Lattice Vibrations of Minerals' Be Lattice Dynamics and an Approximation for Minerals With Application to Simple Substancesand Framework Silicates

SUSAN WERNER KIEFFER x

Departmentof Earth and SpaceSciences, University of California, Los Angeles,California 90024

The principlesof lattice dynamicsare briefly reviewedin this paper, and from theseprinciples a simple, generallyapplicable lattice vibrational spectrumis proposedfor minerals.The spectrumcan be usedto calculatethe thermodynamicfunctions in the harmonic approximation. The model proposedis consistent with lattice dynamicsand is sufficientlydetailed in its assumptionsabout the distribution of modesthat the thermodynamicfunctions are closelyspecified. The primitive unit cell, containings atoms,is chosenas the fundamental vibrating unit; it has 3s degrees of freedom, three of which are acoustic modes. Anisotropy of thesemodes is includedby use of anisotropicshear velocitiesfor the two shear branches: dispersionis includedby useof a sinusoidaldispersion relation betweenfrequency and wavevector for all three acousticbranches. Optic modes,which comprise3s - 3 degreesof freedom, are representedby a uniform continuum,except for 'intramolecular' stretchingmodes (such as Si-O stretchingmodes) which can be enumeratedand identified as being isolated from the optic continuum. Parametersfor the model are obtained from elasticand spectroscopicdata; the model is independentof any calorimetricdata. It is applied to give the temperaturedependence of Cv over the range 0ø-I000øK of halite, periclase,brucite, corundum, spinel, quartz, cristobalite, silica glass,coesite, stishovite, rutlie, albite, and microcline. The specificheat of the simpleDebyelike substances, halite and periclase,is reproducedwell by the model. The influence of the additional formula unit of H•.O on the vibrational and thermodynamic properties of brucite,as comparedto periclase,is discussed.The heat capacitiesof the relativelysimple minerals, spinel and corundum,are given accuratelyby the model. The heat capacitiesof quartz, cristobalite,and coesite are accuratelypredicted from spectroscopicdata through the model. The heat capacity of silica glassis discussedin terms of the classiccontinuous random network (CRN) model and a paracrysta!!inemodel, the pentagonal dodecahedral(PD) model of Robinson (1965). The PD model appears to be more consistentwith measuredCv data than the CRN model. The heat capacity data of rutlie are reasonably reproducedby Ihe model as are the data for stishoviteat temperaturesabove 50øK. Measureddata for stishovitebelow 50øK appear to contain an excessheat capacity relative to the model; this excessmay arise from surfaceenergy contributions, as was suggestedby Holm et al. (1967), and it is suggestedthat the model providesa better estimateof the low-temperaturevibrational heat capacityof stishovitethan the measureddata. The heat capacitiesof albite and microcline are reproducedwell by the model. The model is only a simpleapproximation to real lattice vibrational spectra,but it appearsto work well for a large number of mineralsand is thereforeuseful in correlatingstructural, compositional, elastic, spectroscopic,and thermodynamicproperties for purposesof extrapolationand predictionof theseproperties.

CONTENTS 1. INTRODUCTION In the previoustwo papersof this series[Kieffer, 1979a, b], Introduction ...... 35 Lattice dynamics:Summary of principles...... 36 referredto as paper I and paper 2, respectively,the vibrational Vibrating unit of the crystal...... 36 characteristicsof silicatesas known from simple one-dimen- Modes of vibration of crystals...... 36 sional theory and from experimentaldata have been reviewed, Brief review of methods of calculating g(•0)...... 37 and the influence of various effects such as anisotropy, dis- Formulation of simplifiedmodel for complexsubstances ...... 38 persion,and low- and high-frequencyoptic modeson the heat Cell parameters...... 38 Acoustic modes ...... 38 capacity has been discussedin a general way. In this paper, Optic modes...... 39 severalprinciples which can be used to formulate a generally Summaryof vibrational mode distribution...... 40 applicablemodel for the thermodynamicfunctions of minerals Thermodynamicfunctions ...... 40 are extracted from lattice dynamics theory. The model pro- Data requiredfor model ...... 42 posed lacks the rigor of formal lattice dynamics, but it is Temperaturedependence of the heatcapacity ...... 45 Halite and periclase...... 46 generallyand easilyapplicable to a large variety of substances Brucite ...... 48 of geologic interest. This approach provides a method for Corundum ...... 49 calculatingthe thermodynamicfunctions which is independent Spinel...... 49 of any calorimetric data; the only data required are elastic Quartz, cristobalite,glass, and coesite...... 50 Rutlie and stishovite ...... 53 wave velocities,crystallographic parameters, and spectralfre- Albite and microcline ...... 55 quencies.In this paper the model is illustrated with calcu- Summary...... 56 lations of the temperaturedependence of the heat capacity for Notation ...... 57 the 'simple substances'halite, periclase,brucite, corundum, spinel, stishovite, and rutlie and the framework silicates quartz, cristobalite,coesite, silica glass, albite, and microcline. The effectsof structureand compositionon the lattice vibra- • Now at U.S. Geological Survey, Flagstaff, Arizona 86001. tions and hence on the thermodynamic functions are dis- Copyright¸ 1979 by the AmericanGeophysical Union. cussed.

Papernumber 8R1055. 35 0034-6853/79/018R- 1055506.00 36 KIEFFER:THERMODYNAMICS AND L,•TTICE VIBRATIONS OF MINERALS,3

2. LATTICE DYNAMICS: SUMMARY OF PRINCIPLES For small-amplitudevibrations the potential •I, may be ex- Vibrating Unit of the Crystal panded as a Taylor series in powers of the atomic dis- placementsto terms of secondorder, with the result that the The theory of lattice dynamics is basedon the principle of derivative on the right side of (1) becomes translational invariance of the lattice as expressedin Bloch's theorem: 'For any wave-function/statefunction that satisfies the Schr6dingerequation (or its classicalor quantal equiva- • r r t ? lent) there existsa vector K suchthat translationby a lattice vector a is equivalentto multiplying by the phasefactor exp (iK.a)' [Ziman, 1972, p. 17]. Therefore in lattice dynamicsthe Hence the equationsof motion for particle rnr become fundamentalvibrating unit of the lattice is taken as the Bravais or primitive unit cell [e.g.,Born and Huang, 1954,chapter 5]. If r = rr' r') (3) there are s particlesin the Bravaiscell (where s = Zn, Z is the number of formula units in the cell, and n is the number of There are an infinite number of these equations of motion particles in the formula on which the molecular weight is (involvinga triplesummation over all cellsj'(0 < j' < •), over based), there are 3s degreesof freedom associatedwith the cell all particlesin a unit cell, r' (0 < r' < s - 1), and over three and 3SNA/Z = 3nNA degreesof freedom in one mole of the directions,• (• = 1, 2, 3)). Reductionof this infinite number substance.(Symbols are defined in the notation list.) of simultaneousdifferential equationsto a finite number fol- The Bravais unit cell of some lattices does not coincide with lows from Bloch's theorem that the basic property of lattice the morecommonly used crystallographic cell. Early investiga- periodicityallows wavelike solutions of the form tors in the field of vibrational properties of solids sometimes used a larger unit cell than the Bravaiscell as the vibrating r (m•)•/: r unit. For example, Raman [1943, 1961] formulated a lattice •(j) = 1 W•(r)exp{iK.x(J)_i•t } (4) dynamical theory for sodium chloride in terms of a supercel!, where K is an arbitrary wave vector in reciprocalspace. The the fundamentalBravais cell doubled in length along each above equation is the generalized form of (18) and (25) in dimension. The vibrational modes of such a supercellare paper I for the monatomicand diatomiccases, respectively. In enumerateddifferently from those of a Bravaiscell; the differ- the theoryof vibrationsof elasticcontinua, the magnitudeof y ence between the Raman model and current lattice dynamics = K/2= is proportional to the reciprocalof the wavelengthof models will be discussed in section 5. an elasticwave, and its direction is the directionof the propa- gating wave. W•(r) is the wave amplitude and may dependon Modes of Vibrationof Crystals K or on r througha phasefunction, but it is independentof the Many ...... iormulutlOn:s ' ..... Ol tn<.t_ ..... proo,<• t.• ..... u•e small-amplitude vi- cell indexj. The angular frequencyis w (= 2=e). brations of a lattice are possible[e.g., Born and Huang, 1954; From the assumptionof (4) a set of 3s linear homogeneous Blackman, 1955; Maradudin et al., 1963; Wallace, 1972], but equationsin the 3s unknownsW•(r) (r = 0,1,2,. ß.s - 1; a = the main results are similar in all cases.The following is a l, 2, 3) is obtained(corresponding to (29a) and (29b) in paper summary of the detailed analysisof Born and Huang [1954, I for the diatomic case): chapter 5] and is simply a generalizedform of the discussion presentedin paper I for monatomic and diatomic lattices.The formulation is outlined here to show briefly the argument •:W•(r) = • C•a( rr'K ) w•(r') (5) which leads to a separation of acousticand optic modesand where the coefficientsC• are complicatedfunctions of the their enumeration for polyatomic substances. lattice potentialcoefficients (given by Bornand Huang [1954,p. Consider a perfect crystal lattice. The position vector of a 224]). The 3s equationsare solubleif particle may be representedas •2•a•rr'--Ca•(K)rr' =0 (6) x(•)=x(/') +x(r) whereb•e is the Kroneckerdelta Mnction.The determinanton the left is known as the 'secular determinant' and is a determi- where the positionwithin the unit cell is designatedas x(r) and the distance of the unit cell from the origin is x(/). In these nant equationof 3sth degree in w:. Equation(30) in section3 in expressions,jis the cell index (/= l, 2, 3, ßß ß ), and r is the base paper 1 is the secularequation for the linear diatomic chain. index (r = 1, 2, 3, ...) which distinguishesdifferent nuclei in The 3s solutionsfor each value of K are designatedas w?(K), the cell. The Cartesian componentsof the instantaneousdis- wherei = 1, 2, ... , 3s. It can be provedthat the eigenvalues placementof a particle of massmr from its equilibrium posi- {w?(K)} are the squaresof the normal mode frequenciesof the tion are denoted as crystal: wa(K) is a multivalued function of K and the 3s solutions for each value of K can be regarded as 'branches'of the multivalued function wa(K). If the lattice has a high degreeof symmetry, these branches may coincide for certain values of K. The relations w = wt(K) for 1 • i • 3s are known as the The potential energy of the structure, in terms of the dis- 'dispersion relations.' Dispersion relations for simple mon- placementsof the particles,is denotedby •. The equationsof atomic and diatomicsubstances were discussedin paper 1. For motion of a particle of massrnr are each of the 3s values of wta(K) correspondingto a given K, there exists an eigenvector r = - (1)

whosecomponents are the solutionsto (5). KIEFFER:THERMODYNAMICS AND LATTICEVIBRATIONS OF MINERALS,3 37

The matrixof the coefficientsC•e is known asthe dynamical vector space.Because little is known regarding the nature of matrix. It is Hermitian, and thereforethe eigenvalues{o•?(K)} interatomic forces, the practice has been to assumesimple are real. The microscopiccondition for stability of the lattice is models for forces (e.g., bond-bendingand bond-stretching that the o•?(K) be positive.It follows that o•(K) is either real forces)and to calculatetheir magnitudesfrom the experimen- or purely imaginary. If o•(K) were purely imaginary, vibra- tal valuesof elasticconstants, optical frequencies at K = 0, tional motion of the lattice would erupt exponentiallyinto the and, more recently, inelastic neutron scatteringdata which past or future [Maradudin et al., 1963]. Therefore only real give dispersionrelations in severaldirections in crystals.From o•t(K) representlattice waves. calculated or measureddispersion curves in high-symmetry Enumeration of acoustic and optic modes follows from directions and calculated curves in other directions in the examination of limiting values of K -• 0 [Born and Huang, Brillouin zone in which the normal modes have not been 1954, pp. 230-232] (this is the long-wavelengthlimit at which observed,g(w) is obtained by numericalintegration over the physicalobservations are frequentlymade). In the long-wave- zone volume,a tediousprocess now easedby the useof high- lengthlimit, K is set equalto zero in the coefficientsC• and speedcomputers. This method was introducedby Blackman in (5) for the eigenvectorW(r). As K -• 0, one of the o•'(0) a.nd refined by Houston; Blackman [1955], Leibfried [1955], vanishesfor a = 1, 2, 3 [Bornand Huang, 1954,p. 229]. There- Maradudin et al. [1963], and Cochran[1971] provide detailed fore there are three solutionswhich vanish with vanishingK. reviews of numerical methods of calculation of g(w). In this limit, all particlesin each unit cell move in parallel and For a few simple substances,force modelswhich take into with equal amplitudes.This motion is characteristicof elastic account as many as sixth-neighborinteractions have been deformation under sound waves, and hence these three successfulin predictingdispersion laws for crystalsand thus branches are termed 'acoustic branches.' The remaining 3s the low-temperature behavior of Cu. A sodium chloride - 3 modeswhose frequencies do not vanishat K = 0 are called spectrum calculated by Neubergerand Hatcher [1961] was 'optic modes,'because these modes can interact with photons reproducedas Figure 3 in paper 2 and is usedin section6 for if the particlesbear oppositeelectrical charges. comparisonwith the proposedmodel. This spectrumand an In summary, there are three acousticbranches and 3s - 3 earlier N aCl spectrumcalculated by Kellerman [1940] both optic branchesfor each wave vector K. Dispersionrelations yield good agreementwith measuredheat capacitiesCv or for polyatomicsolids were shownin Figure 5 of paper 2. The calorimetric Debye temperaturesOca•(T) for N aCl. Rauben- exact behaviorof the dispersioncurves is complicatedand it is heimerand Gilat [1967] show that the observed40% decrease therefore not possibleto specifythe shapesof the dispersion in 0cal(T)of zinc (hcp) between0 ø and 15øK is predictable relations without further knowledgeof the interatomic poten- from a force-constantmodel. An effect of similar magnitude tial • and the coefficientsC•. However,the usefulgeneral was calculated for white tin (tetragonal) by Kam and Gilat result which emergesfrom lattice dynamicsis that all of the [1968] from a Born-yon Karman force model. Walker [1956] roots o•i(K) as a function of K are continuousand have the has shown that a consideration of anharmonic effects is nec- periodicity of the reciprocallattice [e.g., Ziman, 1972, p. 36]. essaryto predict O•(T) accuratelyfor aluminum by fitting At this point it should be noted that it is sometimesconve- the Born model of lattice vibrations to experimentalphonon nient and instructive to consider the vibrations of structural dispersioncurves obtained from diffusescattering of X rays. units other than the Bravais unit cell. For example, if a mon- A good fit to the heat capacityat low temperatureswas ob- atomic linear chain is treated as a diatomic chain (two atoms tained for sodiumby Dixon et al. [1963], usingphonon disper- per unit cell) with a massratio approaching1, a spuriousoptic sion relationsobtained by inelasticneutron scatteringexperi- mode is generated. This optic mode is actually the acoustic ments. branch folded back into the artificially small Brillouin zone The calculations of these and many other authors have generatedby the choice of a supercell.Similarly, a diatomic shownthat the lattice dynamicaltheory is essentiallycorrect, linear chain may be treated as a monatomic chain which beinglimited mainly by the lack of informationon interatomic contains new Brillouin zone boundaries across which the fre- forcesand possiblyby considerationsof anharmoniceffects. quenciesare discontinuous.These classicexamples are dis- The theory is powerful becauseit correlatesthermal, elastic, cussedin many elementary texts; the discussionby Brillouin and dielectric propertieswith the fundamental interactions [1953, chapter 4] is particularly good. Such considerationsof amongthe atoms.In practice,the useof this methodinvolves non-Bravaisstructural units are consistentwith lattice dynam- formidable calculations, and it has been used only for rela- ics if the ratio of acousticto optic modes(3/3s) basedon the tively simple crystals. vibrations of a Bravais cell is not altered. Another methodof interest,but asyet limited application,is Fortunately, for calculation of the thermodynamic func- the method of moments[Montroll, 1942, 1943]. In this approx- tions an accuratedescription of the dispersionrelations o•(K) imation the frequencydistribution is expandedin terms of is not required,but rather, a descriptionof the latticevibration momentsof the spectrum,the momentsbeing obtainedfrom spectrumg(o•), and it will prove sufficientfor our purposesto the trace of the characteristicmatrix [Born and Huang, 1954, p. extract the major conclusionsof this section for use in the 72]. This method has successfullyaccounted for thermody- model. Many methodsof calculatingor estimatingg(o•) have namicproperties of somecubic substances. been proposed;the major methodsare reviewedin the next Many authorshave attemptedto find a simplerrepresenta- section. tion of g(o•)than that givenby the latticedynamical theory by usingcombinations of Einsteinand Debye terms.One of the Brief Reviewof Methodsof Calculatingg(o•) earliestapproximations was that of Nernst and Lindemannin The previousdiscussion demonstrates that an accuratecal- 1911 [seeBlackman, 1955], who proposeda combinationof culation of g(o•) requires (1) the knowledge of interatomic two Einsteinoscillators to representthe specificheat according forcesamong large numbersof atoms, (2) the solution of the to equationsof motion for a large number of wave vectors,and (3) the integration of the dispersionrelations over all of wave Cv = (3Nk/2) [E(O/T) + E(O/2T)] (7) 38 KIEFFER: THERMODYNAMICSAND LATTICE VIBRATIONSOF MINERALS, 3

In this equation,8(O/T) is the Einsteinheat capacityfunction. that a complete enumeration of vibrational frequencies This expressioncan adequatelyrepresent the heat capacityof wt•'(K = 0) and their weighting functions is generally not some substancesbut has no strong basis in theory. Similar availablefor substancesof geologicaland geophysicalinterest. modelswhich combine Debye, and Einsteinfunctions have The model proposedin the following sectionis lessrigorous been used to fit and extrapolate thermodynamicdata [e.g., than latticedynamical models based on interatomicforces a•nd Kelley et al., 1953;Kelley, 1929], but the Einsteinfunctions is lessspecific than the spectroscopicmodel, but it is generally usedin suchmodels are obtainedby empiricalfitting and have usefulfor minerals with data currently available. no direct physicalmeaning. Tarasoy[1950] modifiedthe Debye theory to take account 3. FORMULATION OF SIMPLIFIED MODEL of specificstructural features of crystals,such as molecular, FOR COMPLEX SUBSTANCES chain, or layergroups. His work wasextended to a numberof The purposeof this work is to formulate the simplestvibra- highlyanisotropic monatomic lattices (Se, Te, Hg, In, Zn, Cd, tional spectrumthat will predict the thermodynamicfunctions Sb, white Sn, and Li) by De Sorbo [1954]. Newell [1955] of mineralsto an accuracyuseful in geophysicsand petrology. reexamined the theoretical basis of the Tarasov models for In this sectiona model is proposedfor the lattice vibrational lameliarcrystals and found that the predicted behavior (Cv o: spectrumg(w). The model proposedis consistentwith lattice T• at low temperatures)implied specialproperties of the mo- dynamicsand is sufficientlydetailed in its assumptionsabout lecular forcesother than thoseto be expectedsimply from the the distribution of modes that the functionsare closelyspeci- lameliar structure. He consideredit unlikely that a general fied. The model incorporatesthe four effectspostulated in simplesemiphenomenological law couldbe foundfor lameliar paper 1 as causesof the specificheat variationsfrom a Debye crystalscomparable to the generalityof the Debye law for model: anisotropy, dispersion,and low- and high-frequency isotropic materials. optic modes.The model is consistentwith lattice dynamicsin Spectroscopistshave frequently represented complex requiring the use of the primitive unit cell, in correctlyenume- spectraby separatingthe vibrationsof a chosenunit cell or rating acousticmodes, and in recognizinganisotropy and dis- 'molecule' of a substanceinto a discrete'intramolecular' spec- persion of acoustic branches. It is more detailed than the trum of Einstein oscillatorsplus an acousticspectrum of spectroscopicmodel in the descriptionof anisotropyand dis- 'translational' motions of the molecule (as discussedin paper persion in the acoustic branches.It is less rigorous than the 2). The modelis illustratedhere by referringto the calculations spectroscopicmodel becausedetails about the distribution of for quartz by Lord and Morrow [1957]. In the spectroscopic optic modes are ignored. The successof the method results method the assignmentof Einstein oscillatorsto the optic from the insensitivityof the thermodynamicfunctions to de- vibrationsrequires a detailedcompilation of vibrational fre- tails of the spectrumat most frequenciescharacteristic of the quenciesand degeneraciesfrom an assumedlattice model for optic modes. whichthe vibratingunit of the crystalis specified[e.g., Raman, 1943, 1961; Saksena, 1961]. Lord and Morrow [1957] assume Cell Parameters that the fundamentalvibrating unit in quartz is the unit cell, It is a requirement of lattice dynamical theory that the consistingof three SiO: groups.Each unit cell contributes27 primitive unit cell be taken as the fundamentalvibrating unit degreesof freedomto the total for the crystal.They assignthe of the crystal. A molecule of the crystal is here taken as the followingwave numbersw = (w/2•rc) and speciesto the K = 0 unit on which the chemical formula is based. The number of vibrations of quartz observedby infrared and Raman spec- moleculesin a unitcell is designatedby Z,•thenumber of troscopy:species A, 207, 356, 466, and 1082cm-•; speciesB, atoms in a moleculeby n, and the number of particlesin a unit 365, 520, 780, and 1055cm-•; and speciesE, 128,264, 397,452, cell by s; hence s = nZ. There are 3s degreesof freedom 695, 800, 1064,and 1160cm -•. SpeciesE vibrationsare doubly associatedwith the unit cell and 3nNa degreesof freedom degenerate.These frequencies account for 24 degreesof free- associated with one mole of the substance. The unit cell vol- dom and, correspondingly,24 Einsteinoscillators. According ume is designatedas VL and the Brillouin zone volume is to the rationale discussedin the earlier part of this sectionthe therefore (2•r)s/VL. For simplicity the Brillouin zone is re- remaining 3 degreesof freedom are assumedto be trans- placed by a sphere of the same volume with radius Kmax.The lational motions of the unit cell. These 3 degreesof freedom volume of the Brillouin zone, VR, is then obeya Debyefrequency distribution law with a Debyetemper- ature of 254øK, appropriate to one ninth of the degreesof VR= (2•r)S/V•= NA(2;r)S/ZV= i};rKmaxs (Sa) freedom.The Debye temperaturewas determinedfrom experimental heat capacitiesbelow 10øK. The frequencyspectrum Kmax= 2•r(3NA/4•rZV)TM (Sb) constructedin this way (schematicallyillustrated in Figure 3d, where it is to be comparedwith the spectrumproposed in this Equation (8b) definesthe Brillouin zone boundaryKmax and paper) leadsto good agreementbetween observed and calcu- is consistentwith the requirementof the Born theory that the lated heat capacitiesfor quartz except at very low temper- same Brillouin zone be used for all waves (longitudinal and atures,where the specificheat is most senstitiveto detailsof transverse).The shortestwavelength then which can propa- the spectrum. gate through the crystal, Aml,, is• The spectroscopicmodel is basicallya powerfulapproach to modeling the vibrational spectra of minerals and, thereby, •Xmin= 2•r/Kmax= (4•rVZ/3NA)TM (9) calculatingthe thermodynamicfunctions, for it enumerates Acoustic Modes the individual weightedfrequencies w?(K = 0). It doesignore variationsof w•:(K)with K, couplingbetween oscillators, and Three of the 3s degreesof freedommust be acousticmodes. anisotropy and dispersionin the acoustic branches.These Each acousticbranch may be characterizedat long wave- problemslimit its accuracy,especially at low temperatures. lengths(K -• 0) by a directionallyaveraged sound velocity u•, However, the main limitation for minerals is the simple fact u•.(shear branches), or us(longitudinal branch). As was dis- KIEFFER:THERMODYNAMICS AND LATTICE VIBRATIONSOF MINERALS, 3 39

N.• VLNA V cussedin paper1, rigorouscalculation of g(co)for eachacous- dt - - i= 1,2,3 tic branch requiresthe directionalaverage ZVa Z(2•r) s (2•r)s

The acousticspectral distribution is thereforeobtained from u,= •1 fo .q,vf(O, d9 &) i= 1,2,3 (10) (11)-(16) as Directional acousticvelocities v,(O, ok) are generallynot available for minerals;a procedurefor estimatingthe directional g(co)dco= • 3NA(2/;r)a[sin-'(cø/cø')]s averagesfrom partial single-crystaldata or from poly- crystallinedata was givenin the appendixof paper 1. In the modelproposed here the frequencydistribution g(w) where the awkward notation co,.maxis replaced with w• to for the acousticpart of the spectrumis obtained from an denote the maximum frequencyfor each acousticbranch. assumedform for the dispersioncurves for the threeacoustic The assumedsine wave dispersionfor the acousticbranches branches.For simplicity,a simplesinusoidal dispersion rela- causessingularities in g(w) dw at w•, w:, and wa. Becausethe tion is used. This form is rigorously appropriate to linear area under g(w) is finite, the singularitiesin g(w) do not give lattices and representsempirically the approximateform rise to any singularitiesin the thermodynamic functions. found for many of the measuredcurves (see paper 2): Optic Modes Experimentaldata do not yet providedetailed optic mode co,(K)=co,.max(K) sin(Kma•-• K ' '•-•) (11) distributions of most minerals, and theoretical models fail to whereco,.max is the maximumfrequency reached at the zone predictoptic mode distributions accurately because of the lack boundary by branch i. of detailed descriptionsof interatomicforces. However, becausethe thermodynamicfunctions are integralsover the fre- The branch distributionfunctions g,(co) are determinedby quencydistribution, they are insensitiveto detailsof the spec- summing the allowed wave vectorsover the Brillouin zone. If trum. For this reason, in the model proposedhere the optic the vibrationalstates of the crystalcorrespond to wavevectors modesare describedin the simplestway whichsatisfies experi- K whosetips are uniformly distributedin reciprocalspace, the mental observations of the minimum and maximum values of density of vibrational stateshas the form vibrational frequencies;this description in general provides f,(K) dK - 4;rKs dK.d, (12a) adequateaccuracy to describethe thermodynamicfunctions. There are total of 3s - 3 optic modesper unit cell. Although where d, is the density, assumedto be uniform, of vibrational in principle as many as 3s - 3 separateoptic modescould be statesin reciprocalspace. (Compare with Debye theory, (4) in identified and enumerated,in practice,only one or two groups paper 1.) When this equation is expressedin terms of the of modes can be enumerated. They are usually stretching vibrational frequency, it becomes modes,e.g., the internal Si-O stretchingmodes and the AI-O dK stretchingmodes. If the stretchingmodes of one or two types g,(co,)dco,= f,[K(co,)] •-• dco, = a,co? dco, (12b)of molecular groups can be identified, they are representedin the model by weighted Einstein oscillators.(They would be where from (11) most accuratelyrepresented by a band of frequencies,but if single Einstein functions are used for the highest frequency K=•2KmaxSin_l(co,(K) co,,max(K)l.• (13) modes, the errors introduced into heat capacity calculations are small if the modesdo not representmore than about 25% and therefore of the total degreesof freedom. For example, in quartz the difference in heat capacity between a model with a single dK 2 = -- Kmax(CO,max 2 -- CO,S)- i/S (14) Einstein oscillator containing 22% of the modes at 1100 cm-• dco, •r ' and a model with the same number spread over a band be- The slopeof eachacoustic branch is requiredto approachthe tween 1060 and 1172 cm-• is less than 1% in the range 0 ø to acousticsound speedat K = O: 700øK). The model is presentedhere with only one stretching mode in order to avoid notational complexities,but the exten- dco,/dK--} v• as K --, 0 co•• 0 (15) sion to two or more Einstein oscillators is obvious and has been used for some of the minerals studied. The stretching The maximum frequency of each acoustic branch Wl,max, mode is representedby an Einstein oscillator at frequency•, cOs,max,or COS,max is given by and the proportion of vibrational modesat this frequencyis co,,max= otKmax'(2Dr) (16) denoted as q. The density of statesfor the Einstein oscillator will be denotedas g• and can be representedby the Dirac delta where Kmaxis given by (8b). function b: The value of a, in (12b) is determined by the density of vibrational statesin reciprocalspace. As for the Debye theory ge = 3qNAnb(co- (18) discussedin paper 1 it is assumedthat all normal modesof the crystalare representedby wave vectorswithin one reciprocal The remaining3s[1 - (l/s) - q] modesper unit cell or 3s[1- cell becauseonly those vibrational modes are physicallydis- (l/s) - q] nNA modes per mole are assumedto be distributed tinct. Each acoustic branch i accounts for 1/3s of the total uniformly over a range of frequenciesspanning the spectrum degreesof freedomor for 3nNA/3S= NA/Z degreesof freedom betweena lowerlimit cotand an upperlimit cou.This uniformly for a crystalcontaining 1 gram molecularweight of formula distributedband of frequencieswill be calledthe 'optic contin- units. The molar density d, of vibrational statesin reciprocal uum' and will be designatedas go. The densityof statesof this spaceis thus continuum(its height) is evaluatedby requiringthe total num- 40 KIEFFER:THERMODYNAMICS AND LATTICE VIBRATIONS OF MINERALS, 3

ThermodynamicFunctions The total molar specificheat normalizedto the monatomic equivalent,Cv*, then becomes

Cv*=3NAk(2) nZ • '•

',=•• foøø[sin (w?-• (oo/oo,)]:(l•oo/kT): - wa)•/:[exp(l•w/kT) exp(l•w/kT)dw- 1]:

+3NAk( I sl q) f•7'• (w"•'-•C'"-•cl)(•cø/IcT): [expexp(/•/k(t•cø/lcT) T) -dcø 1]:

+ 3N•kq(l•oo•/kT)exp(/•w•/kT) (21) Fig. 1. Schematicmodel vibrational spectrumproposed in this [exp(l•oo•/kT) - 1]: paper for minerals.The singularitiesin the acousticbranches at •0•,•0•., and waare indicatedby three dots on each branch.The dotted curves where N•k = R, the gas constant,and nZ = s, the number of representthe acousticbranches and opticalcontinuum separately. The atoms in the unit cell. solid curve is the total spectrum. This expressioncontains three functionswhich are similarto the Debye function: ber of degreesof freedomfor 1 gram molecularweight of the 'Dispersedacoustic function' crystal to be 3nNA: go = 0 oo< oot oo> $(x•)- • (x?- x:)m(e•'- 1): (22) 3NAn[1-- (l/s)- q] (19a) go= (oo•,-- •) • < • < • 'Optic continuum function' In this representation,•t representsthe lowestextent of the optical modes.As was mentionedin paper 2, this may not be the same frequencyas that observedin Raman or far-infrared x• , (x•,- x•)(ex- 1): (23) spectrataken at K = 0. In fact, the modesat K = Km• are Einstein function more important in thermodynamic calculationsbecause of the K • weighting factor (equation (12a)). Therefore it is assumed •;(xs) = xs• eX•/(eXE- 1)• (24) that the lowest optical mode obeys the following dispersion In theseexpressions, x = •w/kT. $(x•) is the heat capacity law across the Brillouin zone: functionfor a monatomicsolid which has a sine dispersion •(Km•) = •(K = 0)/(1 + mdm•) m (19b) function; where m• and m: are the massesof the heaviestand lightest vibrating units, respectively,at low frequencies.These are xt specifiedin Table 6 of paper 2. is the heat capacity function of a continuum spanningthe range(of nondimensionalizedfrequencies) from xt to x,; 8(xs) Summary of Vibrational Mode Distribution is the heat capacity function of an Einstein oscillator. These The proposedvibrational spectra (shown schematically in functionsare givenin Tables1, 2, and 3. At low temperatures Figure 1) consistof three acousticbranches, an optic contin- the dispersedacoustic function has the samedependence on uum, and an Einsteinoscillator(s) (optional), which are given the third power of the temperatureas the Debye function. by (17), (18), and (19): In termsof thesefunctions the molar heatcapacity, normalized to the monatomicequivalent, is

Cv*- 3NAk •Z •=• $(x•) +3NAk I • - • xz

+ 3NAkqg(x•) (25)

It is most important to emphasizethat the heat capacitycan be calculatedfrom the tables of thesefunctions just as it can be calculatedfrom tables of the Debye or Einsteinfunctions. As an exampleof the useof the tables,consider the heat capacity of quartzat 300øK. Spectraldata give wt = 128cm -•, w, = 809 cm-•, and ws = 1100 cm-•. Mode enumeration gives 0.22 modesin the Si-O stretchingband, representedby an Einstein oscillator at 1100 cm-•. The nondimensionalizedfrequencies are xt = 0.61, x, = 3.88, andxs = 5.28. From (8) and (16) the The asteriskmeans that it is implicit henceforththat go is zero maximum frequenciesof the three acousticbranches are W•,ma• outside of the bounds wt < w < w•. = 103cm -•, W•,ma•= 122cm -•, and W,.m• = 165cm -•, giving KIEFFER:THERMODYNAMICS AND LATTICEVIBRATIONS OF MINERALS,3 41

TABLE 1. DispersedSine Function (22) xz slxzJ O. 1 o. 999323 5.1 0.2q5q35 10.1 0.027960 15.1 o.006660 0.2 0.997321 5.2 0.23q6qO 10.2 0.02695q 15.2 0.00651q 0.3 0.993989 5.3 0.22q270 10.3 0.025993 15.3 0.006373 o.q 0.989350 5.• 0.21•316 10.q 0.025075 15.q 0.006236 0.5 0.983q26 5.5 0.20•767 10.5 0.02q199 15.5 0.006103 O.60. 9762q8 5.6 O.195613 10.6 0.023362 15.6 0.00597q 0.7 0.967853 5.7 0.1868qq 10.7 0.022563 15.7 0.0058q9 O. 80. 95 8282 5.8 0.178qq8 10.8 0.021798 15.8 0.005727 0.9 0.9/47583 5.9 0.170q13 10.9 0.021068 15.9 0.005609 1.0 0.935806 6.0 0.162728 11.0 0.020369 16.0 0.005•9q 1.1 o. 923010 6.1 0.155382 11.1 0.019700 16.1 0.005383 1.2 0.90925q 6.2 0.1q8363 11.2 0.019060 16.2 0.00527q 1.3 0.89q601 6.3 0.1q1659 11.3 0.01Rq48 16.3 0.005169 1 .q o. 879119 6.q 0.135259 11.q 0.017861 16.4 0.005066 1.5 0.862873 6.5 0.129151 11.5 0.017299 16.5 O.00qq66 1.6 0.8q5933 6.6 0.12332q 11.6 0.016761 16.6 O.00qR69 1.7 0.828371 6.7 0.117767 11.7 O. 016245 16.7 0.00q775 1.8 0.810258 6.8 0.112470 11.8 O. 015750 16.8 0.00q683 1.9 0.791662 6.9 0.107q20 I I . q O. 015275 16.9 0.00q593 2.0 0.772656 7.0 0.102609 12.0 0.01q819 17.0 0.004506 2.1 0.753306 7.1 0.098025 12. I O. 01q382 17.1 0.004q21 2.2 0.733681 7.2 0.O93659 12.2 0.013962 17.2 0.00q338 2.3 0.7138q6 7.3 0.089502 12.3 0.013559 17.3 0.00q257 2.q 0.69386q 7.q 0.0855q3 12.q 0.013171 17.q 0.00q179 2.5 0.673795 7.5 0.081775 12.5 0.012799 17.5 0.00q102 2.6 0.653698 7.6 0.078188 12.6 O.012qqO 17.6 0.00q028 2.7 0.633628 7.7 0.07q775 12.7 0.012096 17.7 0.003955 2.8 0.613635 7.8 0.071526 12.8 0.011764 17.8 0.00388q 2.9 0.593769 7.9 0.068q34 12.9 0.01 lqq5 17.9 O.003Rlq 3.o 0.57q074 8.O 0.065q92 13.0 0.011137 18.0 0.0037q7 3.1 0.55q592 8.1 0.062693 13.1 0.0108ql 18.1 0.003681 3.2 0.535361 8.2 0.060029 13.2 0.010556 18.2 0.003616 3.3 0.516q15 8.3 0.057495 13.3 0.010281 18.3 0.003553 3. q O. •97785 8.q 0.055083 13.4 0.010015 18.q 0.003492 3.5 0.q79501 8.5 0.052788 13.5 0.009759 18.5 0.003q32 3.6 0.q61585 8.6 0.05060q 13.6 0.009513 18.6 0.003373 3.? O.qqq060 8.7 0.0q8525 13.7 O. 00927q 18.7 0.003316 3.80.q269q3 8.8 O.046547 13.8 0.00q0qq 18.8 0.003260 3.9 0.q10252 8.90.Oq•66q 13.9 0.008822 18.9 0.003206 •.0 0.393997 9.0 0.0q2871 lq.o 0.008607 19.0 0.003153 •.1 0.378190 9.1 O.Oql16q' lq.1 o.008qoo 19.1 0.003100 q.2 0.362838 9.2 0.039539 lq.2 0.008199 19.2 0.0030q9 q.3 0.3q79q? 9.3 0.037991 lq.3 0.008005 19.3 0.003000 q.q 0.333520 9.a 0.036517 lq.4 0.007R17 lq,q 0.002951 q.5 0.319558 9.5 0.035112 lq.5 0.007636 19.5 0.002903 q.6 0.306061 9.6 0.033773 lq.6 0.007q60 lq.6 0.002857 q.7 0.293028 9.7 0.032q97 lq.7 0.007289 19.7 0.002811 4.8 0.280q54 9.8 0.031281 1•.8 0.007125 19.8 0.002767 q.9 0.268334 9.9 0.030121 lq.9 0.006965 19.9 0.002723 5.0 0.25666q 10.0 0.029015 15.0 0.006810 20.0 0.002680

Xl,maxfor the acousticbranches of 0.49, 0.59, and0.79. Note Entropy, Third law from Table 1 that at room temperaturethe acousticbranches arenearly saturated (x '-• 0) andcontribute nearly the value of theDulong-Petit limit, normalizedto 3/3s degreesof freedom. S* -- Thus the three acoustic branches contribute 0.65 cal mol-' nZ i=l• fox,(Xt [sin-' 2 -- X2(x/x,)]:x)l/ø'(eX --dx 1) øK-' to Cv*. From Table 2 the continuumcontributes (1 - 3/ 27 - 0.22) X 3.90 = 2.63cal mol-' øK-'. FromTable 3 the Einstein oscillator contributes 0.84 X 0.22 = 0.185 cal mol-' nZ • ,:, (x,:- x:)'/: In(1 - e-x) dx øK-'. The total calculated Cv* at 300øK is therefore 3.47 cal mol-' øK-'. For comparison,Cv* from experimentaldata is 3.54 cal mol-' øK-'. + 3Nak1- • - q , (e:- 1) Expressionsfor the entropy,the internalenergy, and the Helmholtzfree energy may be derivedfrom Table 1 in paper 1: Internal energyin excessof that at 0ø.K - 3Nnk1- • - q In(l-e- dx 3NAkT a [sin-' (x/x•)]: x dx E* - nZ (xt: - - 1) + 3Nnkqxr- 3Nnkq In (1 - (27) (•2_..)•]fo x, xø')'/ø'(ex (e%-- 1) + 3NAkT1- • - q , (x,- xt)(ex- l) Helmholtz free energy, above that at OøK

+ 3N.4kTqxr (26) (eXE- 1) nZ •r •_-, (x?- x:) 42 KIEFFER:THERMODYNAMICS AND LATTICE VIBRATIONS OF MINERALS,3

TABLE 2. Continuum Function (23)

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

O. 991 452 0.989251 0.986507 0.983228 0.970056 O. 966567 O. 962543 O. 958002 0.952957 0.947420 0.941407 0.934932 0.928014 0.937303 O. 932629 O. 927454 O. 921787 0.915644 0.909037 0.901984 O. 894502 0.886611 0.878330 O. 895655 O. 889966 O. 883803 O. 877 ! 79 0.870109 O. B62610 O. 854697 0.846392 0.837715 0.828685 0.847944 0.841417 0.834447 0.827051 O. 819242 O. 811 038 O. 802459 O. 793524 0.784254 O. 774670 0.796922 0.789739 0.782147 0.774160 0.765798 0.757077 0.748017 0.738637 0.728961 0.719008 O.744989 O. 737320 O. 729275 O. 720872 0.712127 0.703059 0.693687 O. 684032 O. 674117 O. 663961 0.694023 0.686023 0.677679 0.669009 O. 660031 O. 650765 O. 641228 0.631443 0.621430 0.611211 0.645361 0.637159 0.628645 0.619837 0.610752 0.601410 0.591829 0.582031 0.572036 0.561 865 0.599836 0.591539 0.582961 0.574117 0.565025 0.555705 0.546176 0.536458 0.526570 0.516534 0.557876 0.549571 0.541010 0.532211 0.523190 0.513968 0.504562 O. 494991 O. 485277 O. 475438 0.519609 0.511360 0.5028•0 0.494185 O. 485293 O. 476223 O. 466992 0.457619 0.448124 0.438527 0.484955 0.476810 O. 468456 0.459910 O. •1511 88 O. 442308 O. 433288 O. 4241 47 O. 41 4902 O. 405573 0.453706 0.445701 0.437506 0.429139 0.420614 0.411950 0.403164 O. 394274 O. 385298 O. 376252 o. 425589 0.417747 0.409733 0.401564 0.393254 0.384822 0.376283 O. 367655 O. 358955 O. 350200 o.4oo3o1 0.392637 0.384818 0.376857 0.368773 0.360579 0.352293 O. 343931 O. 335510 O. 327045 o. 377539 0.370062 0.362444 0.354699 0.346843 0.338890 0.330857 O. 322760 O. 31 4614 O. 306435 o.357o14 0.349730 0.342316 0.334787 0.327158 0.319445 0.311661 O. 303824 O. 295947 0.288046 0.338464 0.331372 0.324161 0.316847 0.309443 0.301963 0.294423 0.286838 0.279222 0.271589 o. 321552 0.314750 0.307740 0.300635 0.293450 0.286198 O. 278894 O. 271552 O. 264186 O. 256810 o. 3o6 368 0.299652 0.292838 0.285q37 0.278964 0.271932 O. 26•1856 0.257747 0.250621 0.243491 0.292428 0.285895 0.279270 0.272567 0.265799 0.258979 0.252120 0.245235 0.238338 0.231441 o.279675 0.273318 0.266877 0.260364 0.253792 0.247175 0.240524 0.233853 0.227174 0.220499 o. 267970 0.261783 0.255515 0.249189 0.242805 0.236382 0.229930 O. 223462 O. 21 6990 O. 21 0526 o. 257194 0.251171 0.245076 0.238921 0.232719 0.226481 0.220218 O. 21 3943 O. 207668 O. 201 404 0.24¾244 0.241378 0.235446 0.229459 0.223•129 0.217367 0.211285 0.205194 0.199106 0.193032 0.238030 0.232315 0.226539 0.220713 0.214848 0.208954 0.203044 0.197128 0.191218 0.185323 O. 229475 O. 223996 O. 218279 O. 21 2606 O. 206898 O. 201165 O. 195418 O. 189669 O. 183927 O. 178203 0.221512 0.216081 0.210598 0.205072 O. 199513 O. 193934 0.188343 0.182751 0.177170 0.171608 0.214081 0.208784 0.203438 O. 198051 O. 192636 O. 187203 O. 181 760 0.176320 0.170890 0.165482 0.207132 0.201963 O. 196747 O. 191495 O. 186217 O. 180923 O. 175622 O. 170324 0.165040 0.!59778 0.200620 O. 195572 O. 190483 O. 185359 O. 180211 O. 175050 O. 169883 O. 164723 0.159576 O. 154453 0.194503 0.189573 O. 184604 0.179603 0.174580 0.169546 O. 164508 O. 159478 O. 154462 0.149471 0.188749 0.183931 0.179077 0.174193 0.169290 0.164377 0.159462 0.154556 O. 149665 0.144801 0.183325 0.178616 0.173871 0.169100 0.164311 0.159514 0.154716 0.149929 0.145158 0.14041]] O. 178204 O. 173598 O. 168960 O. 164296 O. 159616 O. 154930 O. 150245 0.145570 0.f40914 0.136284 O. 173361 O. 168855 O. 164318 O. 159757 O. 1551 83 O. 150602 O. 146025 0.141458 0.136910 0.132390 O. 168774 O. 164364 O. 159924 O. 155463 O. 150939 O. 146510 O. 142035 O. 137572 O. 133129 O. 12871 3 O. 164425 0.160106 0.155759 0.151393 0.147015 O. 142634 O. 138258 O. 133894 O. 129550 O. 125234 O. 160293 O. 156062 O. 151 806 O. 147531 O. 143246 O. 138958 O. 134676 O. 130407 O. 126159 O. 121939 0.156364 0.152219 0.148049 0.143861 0.139665 0.135467 0.131275 0.127098 O. 122941 0.118812 0.152624 O. 148560 0.144473 0.140370 0.136259 O. 132147 0.128042 O. 123952 O. 119883 0.115843 O. 149058 O. 145073 O. 141066 O. 137044 O. 133015 O. 128987 O. 124965 O. 120959 O. 116974 O. 113017 O. 145656 O. 141747 O. 137817 O. 133873 O. 129923 O. 125974 o. 122032 0.118107 0.114202 0.110327 0.142406 0.138570 O. 134714 O. 130845 0.126971 0.123099 o. 119234 0.115386 0.111560 0.107762 0.139298 0.135532 O. 131748 0.127952 0.124151 0.120352 o. 116562 0.112788 0.109036 0.105314 0.136324 0.132626 0.128910 0.125184 0.121453 0.117725 0.114007 O. 110305 O. 106625 0.102974 0.133474 O. 129R42 0.126193 0.122533 0.118871 0.115211 o. 111562 0.107929 0.104318 0.100736 0.130742 0.127173 0.123588 0.119993 0.116396 0.112803 o. 10921 9 O. 105653 0.102109 0.098594 0.128119 0.124611 0.121089 0.117557 0.114023 0.110493 O. 106974 O. 103l•71 0.099992 0.096541 0.125601 0.122152 0.118689 0.115217 0.111744 0.108276 O. 104818 O. 101378 0.097960 0.094572 0.123180 0.119788 0.116383 0.112970 0.109556 0.106147 O. 102749 O. 099368 O. 096010 O. 092681 0.120851 0.117514 0.114165 0.110808 0.107451 0.104100 o. 10075 9 0.097436 0.094136 0.090865 0.118609 0.115326 0.112031 0.108728 0.105426 0.102130 0.09R845 O. 095578 O. 092334 O. 089118 0.116450 0.113218 0.109975 O. 106725 0.103476 0.100234 0.o970o3 0.093790 0.090599 0.087437 0.114367 0.111185 0.107993 0.104795 0.101597 0.098407 0. 095228 O. 092067 O. 088929 O. 085819 0.112359 9.109225 0.106082 0.102933 0.099785 0.096645 0.093516 0.090406 0.087318 0.084259 0.110420 0.107333 0.104237 0.101136 0.09•037 0.094945 O. 091865 O. 088804 O. 085765 O. 082755 O. 108547 O. 105505 0.102455 0.099401 0.096349 0.093304 O. 090271 O. 087258 O. 084266 O. 081 303 0.106736 O. 103739 O. 100733 0.0977211 0.09ll717 0.091718 0.088732 0.085764 0.082818 0.079901

(spectraldata are given in Table 5); and (3) frequenciesand _ 3 partitioningfractions for any intramolecularmodes separated 3NAkT(I -•-j- q) from the optic continuum (Table 5). (x.- xt) l In(1 - e- dx The model is demonstratedin this paper with calculationsof the temperature dependenceof the heat capacity for several + 3NAkrq In (1 -- e-%) (28) simplesubstances and framework silicates;chain silicatesand orthosilicateswill be considered in paper 4 (S. W. Kieffer, unpublishedmanuscript, 1979). The temperatureand pressure 4. DATA REQUIREDFOR MODEL dependenceof the internal energy,heat capacity,entropy, and Helmholtz free energywill be given in paper 5 (S. W. Kieffer, The modelproposed in section3 allowscalculation of the unpublishedmanuscript, 1979). thermodynamicfunctions from the followingacoustic and Specificheat data for mineralsat atmosphericpressure are spectroscopicdata: (1) threeacoustic velocities u•, u2, and us usuallymeasured and tabulated in terms of C•,, the specific from whichthe acousticcutoff frequencies w•, w2,and ws(or heat at constant pressure, rather than in terms of Cv, the w•,w•, andws) are calculated from (8) and(16) (acousticdata quantitycalculated by (25). C•, and Cv are relatedby aregiven in Table4); (2) twospectral frequencies wtand w, (or wt and w,) whichdefine the limitsof the opticcontinuum Cv = Cv- TI&•B (29) KIEFFER:THERMODYNAMICS AND LATTICE VIBRATIONSOF MINERALS, 3 43

TABLE 2. (continued) 1.1 1.2 1.3 I .4 1.5 1.6 1.7 1.8 1.9 2.0

0.869681 0.850685 O. 851365 O. 841745 0.735341 0.819325 0.809657 0.799705 0.789491 0.779041 0.768377 0.757523 0.746504 0.678090 0.666639 0.764794 0.754651 0.744261 0.733650 0.722840 0.711855 0.700721 0.689458 0.620786 0.609331 0.708802 0.698364 0.687720 0.676891 0.665q01 0.654773 0.643529 0.632193 0.653587 0.643018 0.632277 0.621386 0.610368 0.599246 0.588041 0.576775 0.565469 0.554145 0.600807 0.590240 0.579534 0.568708 0.557787 0.546791 0.535741 0.524658 0.513562 0.502473 0.551540 0.541082 0.530512 0.519851 0.509121 0.498343 0.487536 0.476720 0.465915 0.455137 0.506370 0.496099 O. 485741 0.475318 0.464849 0.454353 0.443851 0.433360 0.422898 0.412484 O. 4654 95 O. 455468 O. q45376 O. 435240 0.1125077 0.414907 0.404749 0.394618 0.384534 0.374511 O. 428845 O. ql 9099 O. 409307 O. 399488 0. 389660 0. 379841 O. 370048 O. 360297 O. 350605 O. 340987 O. 396177 O. 386734 O. 377261 O. 367776 O. 358297 O. 348840 0.339421 0.330056 0.320761 0.311548 0.367156 0.358027 0.348881 0.339736 0.330609 0.321515 O. 312469 O. 303486 O.294581 O.285766 0.341407 0.332594 0.323776 0.314969 0.306189 0.297452 0.288771 0.280161 0.271634 0.263203 o. 318554 0.310053 0.301556 O. 293080 0.284639 O. 276247 0.267,919 0.259667 0.251503 0.243439 0.298239 0.290042 O. 281857 O. 273700 O. 265585 O. 257526 0.249534 0.241624 0.233805 0.226089 0.280136 0.272232 O. 264347 O. 256497 O. 248694 O. 240950 0.233279 0.225692 0.218199 0.210812 O. 263954 O. 256331 O. 248733 O. 2411 75 O. 233668 O. 226224 0.218856 0.211574 0.204389 0.197309 O. 249438 0.242083 0.234759 0.227477 0.220250 0.213090 0.206008 0.199013 0.192116 0.185326 0.236369 0.229269 0.222203 0.215184 0.208222 0.201329 O. 194515 0.187791 0.181164 O. 174644 O. 224557 0.217699 0.210878 0.204106 0.197394 0.190753 0.184192 0.177720 0.171347 0.165081 O. 21 3841 O. 207212 0.200622 0.194084 0.187607 0.181203 O. 174879 0.168645 0.162509 0.156479 0.204082 0.197669 0.191299 0.184981 O. 178726 0.172543 0.166443 O. 160432 0.154518 0.148710 O. 195163 O. 188955 0.182791 0.176681 O. 170635 0.164662 O. 158770 O. 152969 O. 147264 O. 141663 O. 186983 O. 180968 0.175000 O. 169086 0.163237 0.157461 0.151766 0.146161 0.140652 0.135246 O. 179456 O. 173625 O. 167840 O. 16211 2 O. 156448 O. 150858 O. 145349 O. 139928 O. 134603 O. 129379 0.172508 0.166850 0.161240 O. 155687 O. 150198 O. 144783 0.139448 0.134201 0.129049 0.123997 0.166076 0.160583 0.155138 0.149750 0.144426 0.139177 O. 134006 O. 128924 O. 123933 O. 119042 0.160105 0.154768 0.149479 0.144247 0.139081 0.133937 O. 128972 0.124043 0.119206 0.114467 0.154548 0.149358 0.144218 0.139134 0.134115 0.129169 0.124301 0.119518 0.114826 0.110229 0.149363 0.144313 0.139314 0.134371 0.129492 0.124685 0.119956 0.115311 0.110755 0.106294 0.144514 0.139598 0.134732 0.129922 0.125176 0.120502 0.115904 0.111390 0.106963 0.102629 0.139970 0.135181 0.130441 0.125758 0.121139 0.116590 O. 11211 7 O. 107726 O. 103422 O. 099209 0.135703 0.131034 0.126416 0.121853 0.117354 0.112924 O. 108559 O. 104296 O. 100107 0.096009 0.131688 0.127135 0.122631 0.118183 0.113798 0.109481 O. 105239 O. 101077 0.096999 0.093009 0.127904 O. 123460 0.119056 0.114727 0.110451 0.106242 O. 102107 0.098051 0.094077 0.090191 0.124332 0.119993 0.115703 0.111468 0.107295 0.103190 0.099156 0.095201 0.091326 0.087538 0.120953 0.116714 0.112525 0.108389 0.104315 0.100307 0;096371 0.092511 0.088732 0.085037 0.117754 0.113610 0.109516 0.105476 0.101496 0.097582 0.093738 0.089970 0.086281 0.082675 0.114719 0.110667 0.106664 0.102715 0.098825 0.095000 0.091245 0.087564 0.053962 0.080441 0.111837 0.107873 0.103957 0.100095 0.096291 0.092552 0.088881 0.085284 0.081764 0.078324 O. 109096 O. 105216 O. 101385 0.097605 0.093884 0.090227 0.086637 0.083120 0.079678 0.076316 0.106487 0.102688 0.098936 0.095236 0.091595 0.088016 0.084503 0.081063 0.077696 0.074408 0.103999 0.100278 0.096603 0.092980 0.089414 0.085910 0.082472 0.079105 0.075810 0.072593 0.080536 0.077239 0.074014 0.070865 O. 101626 0.097978 0.094378 0.090828 0.087335 0.083903 0.069217 0.099358 0.095782 0.092252 0.088773 0.085350 0.081988 O. 078690 O. 075460 O. 072301 O. 0?6925 O. 0?3?60 O. 0?0665 0.067644 0.097189 0.093682 0.090221 0.086810 0.083454 0.080158 0.066141 0.095113 0.091672 0.088277 0.084931 0.081640 0.078408 0.0?5239 0.072136 0.069102 0.093124 0.089747 0.086415 0.083132 0.079904 0.076733 O. 0?3624 O. 070581 O. 067606 O. 064793 0.091217 0.087901 0.084630 0.081408 0.078239 0.075128 o. 072078 0.069092 0.066174 0.063326 0.089386 0.086129 0.082917 0.079754 O. 076643 0.073589 0.070595 0.067665 0.064801 0.062007 0.087627 0.084428 0.081273 0.07•165 0.075110 0.072111 0.069171 O. 066295 O. 063484 O. 060742 0.085936 0.082792 0.079692 0.076639 0.073637 0.070692 0.067804 0.064980 0.062219 0.059527 0.084309 0.081218 0.078171 0.075171 0.072221 0.069327 0.066490 0.063715 0.061004 0.058359 O. 082742 O. 079703 O. 076707 O. 073758 O. 070859 O. 068014 O. O65226 O. 062499 0.059835 O. 057237 O. 081233 O.078244 O.075297 O. 072397 O. 069546 O.066749 0.064009 O. 061 329 O. 0587 10 O. 056156 0.079777 0.076836 0.07393R 0.071085 0.068281 0.065531 0.062836 0.060201 0.057626 0.055116 O.078372 O.075479 O.072627 O.069820 O.067062 O.064356 0.061705 0.059114 0.056582 0.054113 0.060615 0.058065 0.055574 0.053146

where• is the volumecoefficient of thermalexpansion, B is the temperatures(•700øK) wherethe Cp-Cvcorrection is large(a bulk modulus,and V is the specificvolume: •, B, and V in few percent),anharmonic effects that are not consideredin the generaldepend on the temperature. They were specified asa modelmay be more importantthan the Cp-Cv correction(see functionof temperatureT as follows: discussionbelow). (Cp - Cv)/Cp is generallyless than 1% at 100øK and is about 4% at 700øK for silicate minerals. For • = •o + •T+•,T • halite the correction from Cp to Cv is large (•5%)even at e = eo + (aeo/• T)AT (303 room temperature;values of Cv for halite havebeen obtained from Cp data of Raman [1961] and Janaft [1965a, b, 1966, V = Vo + (aVo/aT)AT 1967]. whereB0 and V0 are the values at 298øK. Values of theparam- For comparisonwith the predictionsof the harmonictheory eterswere obtained from Clark [1966] or estimatedfrom struc- developedhere, Cv thus obtainedfrom Cv must be corrected turalanalogues. The correction from Cp to Cvis significant to a fixed volume, e.g., the volume of the crystal at 0øK, onlyat hightemperatures (T > 300øK)for most minerals. because the theory has assumedthat the effectsof thermal (Barronand Munn [1968] have pointed out that Ce - TVe'Bis expansioncan be neglected,i.e., that volumeis not a function nota simplequantity for nonisotropic solids. It is actually the of temperatureand that the lattice vibrational spectrumis heatcapacity defined under the double restraint of constanttherefore not a function of temperature. In fact, thermal ex- volumeand isotropic stress. Corrections from this quantity to pansionhas the effect of shiftingthe spectrum,to lower fre- theheat capacity atconstant strain are of the order of 1%.) At quenciesif the materialexpands as the temperatureis raised. 44 KIEFFER:THERMODYNAMICS ANDLATTICE VIBRATIONS OFMINERALS, 3

TABLE 2. (continued)

2.1 2.2 2.3 2.q 2.5 2.6 2.7 2.8 2.9 3.0

0.655128 O. 643578 0.632008 0.620439 O. 597847 0.586356 0.574876 0.563425 O. 55'2023 0.540684 O. 529425 O. 51 8261 O. 507204 0.542820 0.531516 O. 520249 O. 509037 O. 497897 O. 486844 O. 475892 O. 465054 O. 454343 O. 443771 0.491410 O. 480390 O. 469429 O. 458545 O. 447752 O. 437064 O. 426495 O. 416057 O. 405761 O. 395616 O. 444407 O. 433738 0.423148 0.412652 0.402262 0.391993 0.381856 0.371862 0.362021 0.352342 O. 4021 32 O. 391 859 0.381679 0.371606 0.361653 0.351831 0.342151 0.332624 0.323257 0.314060 O. 364565 O. 354710 0.344959 0.335326 0.325823 0.316459 0.307244 0.298188 0.289299 0.280583 O. 331456 O. 322027 0.312711 0.303520 0.294465 0.285556 0.276802 0.268209 0.259787 0.251539 0.302431 0.293423 0.284535 0.275778 0.267161 0.258594 0.250385 0.242239 0.234265 0.226467 0.277053 O. 268455 0.259981 0.251642 0.243446 0.235403 0.227518 0.219795 0.212248 0.204874 O. 254879 O. 246673 0.238595 0.230654 0.222859 0.215216 0.207732 0.200412 0.193262 0.186285 0.235485 O. 227653 O. 21 9950 O. 212385 O. 204966 O. 197699 O. 190590 O. 183645 O. 176867 O. 170260 0.218486 0.211005 0.203655 O. 196443 O. 189376 0.182461 O. 175702 O. 169105 O. 162673 0.156409 0.203538 0.196388 0.189368 O. 182486 O. 175748 0.169160 O. 162727 O. 156452 O. 150339 0.144392 0.190345 0.183503 0.176792 0.170217 0.163786 0.157502 0.151370 0.145394 0.139577 0.133921 O. 178651 O. 172098 0.165674 O. 159386 O. 153238 0.147237 O. 141384 O. 135685 O. 130141 0.124754 O. 168239 O. 161956 0.155800 0.149775 0.143895 0.138154 0.132561 0.127117 0.121825 0.116686 O. 158928 O. 152895 0.146989 0.141215 0.135577 0.130080 0.124726 0.119518 0.114459 9.109549 O. 150562 O. 144764 0.139091 0.133547 0.128137 0.122865 0.117733 0.112744 0.107900 0.103203 0.143013 0.137434 0.131978 0.126649 0.121451 0.116388 0.111462 0.106677 0.102032 0.097530 O. 136173 O. 130798 0.125544 0.120415 O. 115414 0.110546 O. 105812 0.101215 0.096756 0.092435 O. 129948 O. 124754 0.119699 0.114757 0.109941 0.105254 0.100699 0.096277 0.091989 0.087837 O. 124262 O. 119258 0.114369 0.109602 0.104958 0.100441 0.096051 0.091793 0.087665 0.083669 0.119050 0.114213 0.109491 0.104887 O. 100403 0.096044 0.091810 0.087704 0.083725 0.079875 O. 114255 O. 109576 O. 105009 O. 100558 0.096226 0.092014 0.087926 0.083961 0.080122 0.076408 0.109830 O. 105299 O. 100878 0.096571 0.092380 0.088307 0.084355 0.080524 0.076815 0.073228 0.105734 0.101342 0.097059 0.092887 0.088829 0.084887 0.081062 0.077356 0.073769 0.070301 O. 101932 0.097672 0.093518 0.0R9473 0.085540 0.081721 0.078016 0.074427 0.070955 0.067599 0.098393 0.094257 0.090226 0.086302 0.082486 O. 07,4782 0.075190 0.071712 0.068347 0.065096 O. 095092 0.091073 0.087158 0.083347 0.079642 0.076047 0.072562 0.069187 0.065924 0.062772 O. 092005 O. 088098 0.084291 0.080587 0.076988 0.073496 0.070110 0.066834 0.063666 0.060608 0.089112 0.085310 0.081607 0.078004 0.074505 0.071110 0.067820 0.064636 0.061559 0.058587 0.086395 0.082693 0.079088 0.075582 0.072177 0.068873 0.065673 0.062578 0.059586 0.056698 0.083839 0.080232 0.076720 0.073305 0.069989 0.066774 0.063659 0.060646 0.057735 0.054926 O. 081 430 O. 07791 3 0.074490 0.071162 0.067931 0.064798 0.061764 0.058831 0.055996 0.053262 O. 079155 O. 075725 0.072386 0.069140 0.065990 0.062936 0.059979 0.057120 0.054359 0.051695 0.077004 0.073656 0.070397 0.067230 0.064157 0.061178 0.058295 0.055507 0.052815 0.050218 0.074967 0.071697 0.068515 0.065423 0.062423 0.059516 0.056702 0.053982 0.051356 0.048823 0.073035 O. 069840 0.066731 0.063711 0.060781 0.057941 0.0551911 0.052539 0.049975 0.047504 0.071200 0.068076 0.065037 0.062086 0.059222 0.056448 0.053764 0.051171 0.048667 0.046254 0.069455 0.066399 O. 063428 O. 060541 O. 057742 O. 055030 O. 052406 O. 049872 O. 047425 O. 045068 0.067794 0.064803 0.061896 0.059072 0.056333 0.053681 0.051116 0.048637 0.046246 0.043941 0.066210 0.063282 0.060436 0.057672 0.054992 0.052397 0.049•47 0.047462 0.045123 0.042869 0.064698 0.061831 0.059043 0.056337 0.053713 0.051173 0.048716 0.046343 0.044054 0.041848 O. 063254 O. 060444 0.057714 0.055062 0.052492 0.050004 0.047598 0.045275 0.043034 0.040875 0.061873 0.059119 0.056442 0.053844 0.051326 0.048889 0.046531 0.044255 0.042060 0.039946 0.060551 0.057850 0.055226 0.052679 0.050210 0.047821 0.045511 0.043281 0.041130 0.039058 O. 059284 O. 056635 0.054061 0.051563 0.049142 0.046799 0.044534 0.042348 0.040240 0.038209 O. 058069 O. 055469 0.052944 0.050493 0.048118 0.045820 0.043598 0.041454 0.039387 0.037396 0.056903 0.054351 0.051872 0.049466 0.047136 0.044880 0.042701 0.040598 0.038570 0.036617 O. 055783 O. 053277 0.050842 0.048481 0.046193 0.043979 0.041840 0.039776 0.037786 0.035870 O. 054706 O. 052244 0.049•53 0.047533 0.045287 0.043113 0.041013 0.038986 0.037033 0.035152 0.053669 0.051250 O. 048901 0.046623 0.044416 0.042281 0.040218 0.038228 0.036309 0.034463 O. 052671 O. 050294 0.047985 0.045746 0.043577 0.041480 0.039453 0.037498 0.035613 0.033800 0.051710 0.049372 0.047103 0.044901 0.042770 0.040708 0.038717 0.036795 0.034944 0.033162 0.050783 0.048484 0.046252 0.044088 0.041992 0.039965 0.038007 0.036119 0.034299 0.032547

An approximatecorrection for theeffect of thermalexpansion discussthe effectof anharmonicityand surfacecontributions can be madethrough the formula[Barron et al., 1959] for MgO;Leibfried and Ludwig [ 1961] discussanharmonicity Ocal(•/rO)/Ocal(•/rT) --- (Po/PT (31') in detail.Anharmonicity isprobably the most important of the effectsignored because it becomesnoticeable at hightemper- wherep0 and pr are the densitiesof the crystalat 0ø and TøK 'aturesof interestto geologistsand geophysicists.For several and3• is thethermal Griineisen parameter. Using this formula, mineralsthere is a tendencyfor 0ca•(T)at hightemperatures to Barronet al. [1959]have shown that the effectis verysmall for decreasewith increasingtemperature (see Figures 2-4). The MgO, being0.5% at 270øK. The effectis expectedto be of same behavior has been observedfor some of the alkali halides similar magnitude for most of the minerals consideredhere andinterpreted as an anharmoniceffect [Barron et al., 1959] and henceis neglected. becausethe 'harmonic'contribution to the heat capacity The model gives the contribution of the harmonic lattice wouldbe expectedto giveconstant Oca•(T) at thesetemper- vibrationsto Cv: Schottkyanomalies, electronic or magnetic atures. contributions,defect, domain, or surfacecontributions, and As wasmentioned above, the effectof thermalexpansion anharmoniceffects have been ignored.Their contributionto alonewould be to causethe spectral frequencies to decreaseas theheat capacity is expected to besmall except possibly at very the temperatureincreases. Thus 0c,•(co), which would ap- low (<10øK) or very high (>500ø-1000øK)temperatures. proacha constantvalue for a harmonicspectrum, would de- Gopal[1966] reviews most of theseeffects; Barron et al. [1959] creaseif thermal expansionalone were considered.Barron et KIEFFER:THERMODYNAMICS AND LATTICE VIBRATIONS OF MINERALS, 3 45 ai. [1959]have calculated the magnitudeof this effectfor a in thispaper in orderto illustratethe effect of compositionand numberof halides(including sodium chloride) which have low structure on the vibrational and thermodynamicproperties. Debye temperatures.Although the expansiondoes cause a Halite, periclase,brucite, corundum, and spinel are considered decreasein 0½a•(oo.),it cannot fully 'explainthe magnitudesof as examplesof simplesubstances with varyingdegrees of ani- the observeddecreases in 0ca,(øø)for the halides.Barron et ai. sotropyand complexity;the five silica polymorphs,quartz, [1959] attributethe excessdecrease to an increasein heat cristobalite,glass, coesite, and stishovite,are examinedas capacitydue to anharmonicityof the latticevibrations them- isochemicalpolymorphs with structuralvariations; futile is selves.At high temperaturesthe mean amplitudeof atomic consideredas an anisotropicmaterial and as an analogueof vibrationshoenrnoe large in enmpari.qnntn the interatomic stishovite;and albite and microcline are consideredas two distances,and cubicand quarticterms in the atomicpotential typicalfeldspars. Data usedin the modelare given in Tables4 (neglectedin the harmonictheory) become important. These and 5. cubicand quartic terms may give rise to positiveand negative The results of the calculations of Cv with the model are contributions,respectively, to the anharmonicheat capacity. givenin Figures2a, 3a, and4a, wherethey are compared with The balancingof thesetwo termsdepends on detailsof the experimentaldata. Curvesof 0½•,(T),which are a moresensi- interatomic forces. tive indicator of deviations from the measured values than are Cv, calculatedfrom (29) and(303, and O½•,(T) obtained from the specificheat curvesand which clearly show systematic thesedata are shownas dots in Figures2-4 for the minerals deviations of the model from the data, are shown in Figures considered here. Data were obtained from references listed in 2b, 3b, and 4b. Discrepanciesbetween the modeland the data the caption of Figure 4 in paper 1. below 100øK are stronglyemphasized by the 0½•(T) curves. The relative contributions of the acoustic,optic continuum and intramolecularmodes are given in Figures2c, 3c, and 4c. 5. TEMPERATURE DEPENDENCE The model frequencydistributions are shownin Figures2d, OF THE HEAT CAPACITY 3d, and 4d, where they are comparedwith spectraldata. Be- The temperaturedependence of the heatcapacity of a num- causethe complete frequency distribution g(v) is not available ber of simplesubstances and frameworksilicates is examined for most minerals,only infrared and/or Raman data can be

TABLE 3. Eins.•i•nFunction (24)

xE E(xEl XE E(xEl 'o.1' 0.999161 5.1 0.1b0528 10.1 0.004191 r 15. I O. 000063 O.2 0.996670 5.2 0.150827 10.2 0.003867 15.2 0.000058 O.3 O.992535 5.3 0.141624 10.3 0.003568 15.3 0.000053 O.q 0.986771 5.# 0.132901 10.q 0.003292 15.4 O. 000049 O.5 0.97942• 5.5 0.124641 10.5 0.003036 15.5 O. 000045 O.6 0.970532 5.6 0.116827 10.6 0.002800 15.6 O. 000041 O.7 0.9601•8 5.7 0.109442 10.7 0.002581 15.7 O. 000037 O.8 0.9•8331 5.8 0.102466 10.8 0.002379 15.8 O. 000034 O.9 O.935148 5.9 0.095885 10.9 0.002193 15.9 O. 000031 1.0 O.92O674 6.O 0.089679 11.0 0.002021 16.0 O. 000029 1.1 0.90•985 6.1 0.083833 11.1 0.001862 16.1 O. 000026 1.2 0.888170 6.2 0.078329 11.2 0.001715 16.2 O. 000024 1.3 0.870314 6.3 0.073151 11.3 0.001580 16.3 0.000022 1.# 0.851508 6.# 0.068284 11.q 0.001455 16.•1 O. 000020 1.5 0.8318#8 6.5 0.063712 11.5 0.001340 16.5 0.000019 1.6 0.811#29 6.6 0.059419 11.6 0.001233 16.6 0.000017 1.7 0.790345 6.7 0.055392 11.7 0.001135 16.7 0.000016 1.8 0.768693 6.8 0.051616 11.8 0.001045 16.8 0.000014 1.9 •.746567 6.9 O.O48O78 11.9 0.000962 16.9 O. 000013 2.0 0.72•061 7.0 0.044764 12.0 0.000885 17.0 0.000012 2.1 0.701266 7.1 0.041662 1•.1 0.000814 17.1 0.000011 2.2 0.6?8268 7.2 0.038761 12.2 0.000749 17.2 0.000010 2.3 0.65515.4 7.3 O.O36O48 12.3 0.000689 17.3 0.000009 2.# 0.632002 7.4 0.033513 12.4 0.000633 17.4 0.000008 2.5 0.608890 7.5 0.031145 12.5 0.000582 17.5 0.000008 2.6 0.585890 7.6 0.028935 12.6 0.000535 17.6 0.000007 2.7 0.563067 7.7 0.026872 12.7 0.000492 17.7 0.000006 2.8 0.540486 7.8 0.024949 12.8 0.000452 17.8 0.000006 2.9 0.518203 7.9 0.023155 12.9 0.000416 17.9 0.000005 3.0 0.496269 8.0 0.021484 13.0 0.000382 18.0 0.000005 3.1 0.474732 8.1 0.019927 13.1 0.000351 18.1 0.000005 3.2 0.053633 8.2 0.018478 13.2 0.000322 18.2 O.00000q 3.3 0.433010 8.3 0.017129 13.3 0.000296 18.3 O.00000q 3.4 0.412894 8.4 0.015874 13.q 0.000272 18.4 0.000003 3.5 O. 393313 8.5 0.014707 13.5 0.000250 18.5 0.000003 3.6. 0.374290 8.6 0.013621 13.6 0.000229 18.6 0.000003 3.7 0.355843 8.7 0.012613 13.7 0.000211 18.7 0.000003 3.8 0.337987 8.8 0.011676 13.8 0.000193 18.8 0.000002 3.9 0.320732 8.9 0.010806 13.9 0.000178 18.9 0.000002 4.0 0.304087 9.O 0.009999 lq.0 0.000163 19.0 0.000002 4.1 0.288055 9.1 0.009249 14.1 0.000150 19.1 0.000002 4.2 0.272637 9.2 0.008554 14.2 0.000137 19.2 0.000002 q.3 0.257832 9.3 0.007909 14.3 0.000126 19.3 0.000002 q.q 0.243635 9.4 0.007311 14.4 0.000116 19.4 0.000001 q.5 O.23OO40 9.5 0.006756 14.5 0.000106 19.5 0.000001 q.6 0.217038 9.6 0.006243 14.6 0.000097 19.6 0.000001 4.7 0.204620 9.7 0.005767 14.7 0.000089 19.7 0.000001 4.8 0.192773 9.8 0.005326 lq.8 0.000082 19.8 0.000001 4.9 0.181485 9.9 0.004918 lq.9 0.000075 19.9 0.000001 5.0 0.170742 10.0 O.00qSqO 15.0 0.000069 20.0 0.000001 46 KIEFFER:THERMODYNAMICS ANDLATTICE VIBRATIONS OFMINERALS, 3

TABLE 4. Data for AcousticBranches: Simple Minerals and FrameworkSilicates

Commonly Designated Crystallo- Z Vu,it cell graphic (Primitive ( Mineral Structure Cell) n 10-•-4cm3 gcm -3 kms-• kms-• kms-• kms-• kms-• øK cm-t cm-• cm-•

Halite fcca 1 2 45 2.16 2.500 2.74 2.61 4.56 2.90 306 93 102 170 Periclase fcca 1 2 19 3.58 5.830 6.30 6.05 9.71 6.66 942 289 312 482 Brucite trig 1 5 64 2.37 4.25c 4.25 (4.25) 7.36 4,72 697 141 141 244 Corundum rhom(hex)" 2 5 85 3.99 6.120 6.59 6.41 10.85 7.03 1026 184 198 327 Spinel fcc" 2 7 132 3.58 5.310 6.00 5.62 9.82 6.24 879 138 156 255 a-Quartz trig 3 3 113 2.65 3.760 4.46 4.05 6.05 4.42 567 103 122 165 a-Cristobalite tetr 4 3 171 2.32 (3.56)d (4.23) (3.84) (6.17) 4.23 519 85 101 147 Silicaglass seetext seetext seetext seetext 2.20 3.74 3.74 3.74 5.50 4.08 491 seetext see text see text Coesite mono" 8 3 274 2.92 4.170 5.24 4.59 8.19 5.10 676 85 107 167 Stishovite tetr 2 3 47 4.20 (5.05)e (6.16) (5.50) 11.00 6.16 921 186 227 405 Rutlie tetr 2 3 62 4.26 4.730 5.73 5.14 9.26 5.72 781 158 191 309 Albite tri 4 13 665 2.62 2.9803.91 3.33 6.06 3.70 472 45 59 92 Microcline tri 4 13 722 2.56 3.0203.85 3.34 6.02 3.72 460 45 57 89

"Forthese substances the primitive cell differs from the commonly designated crystallographic cell. øReferencesfor datagiven in TableAI of paper1. CAnisotropynot includedbecause of lack of data (seetext). dAssumedVva..s/p = 1.62;Vva..•,/p = 2.65[after Anderson and Liebermann, 1966]. Assumed u2/vva.,s = 1.10,as for quartz. eValuesof Vva. fromMizutani et al. [1972];assumed UdVva.,s = 1.12,as for rutlie. shown for many minerals. It should be rememberedthat the arbitrarily as a free parameter,for example,to match an frequencydistribution inferred from suchspectroscopic data observedheat capacity at 298øK. The onlytemperature range taken at K = 0 is incomplete becausedispersion curves are where a single-parameterDebye model might be usedwith not known over the whole Brillouin zone. confidenceis at high temperatureswhere 0•(co) is ap- I n all casesthe heatcapacities given by the modeldeveloped proached.In this regiona Debye(or an Einstein)model with here are in much better agreementwith the measuredheat 0o (or 0E) equal to 0cal(cO)might be appropriate. capacitiesthan are heat capacitiescalculated from the Debye modelwith 0o basedon 0e•.(Such a modelwould give straight Halite and Periclase linesat the 0øK valueof Oca•(T)marked by an asteriskin the Halite (NaCI) and periclase(MgO) (seeFigure 2) haveone curvesof Figures2b, 3b, and 4b.) For simplesubstances (hal- formulaunit in the fundamentalBravais unit cell.They thus ite, periclase,spinel, and corundum)the heat capacitypre- have six vibrational modes, three of which are acoustic.The dictedby the modelexhibits nearly Debyelike behavior in that mode partitioning obtained from models which choose a Oca•(T)attains a constantvalue at relativelylow temperatures. primitiveunit cell as the fundamentalvibrating unit is very The heatcapacity of the more complexsubstances is not at all differentfrom the mode partitioningobtained from models Debyelike.For thesesubstances the single-parameter Debye whichchoose other vibrating units; for example, Raman [1943, modelis quite inadequateto predictthe specificheat. This 1961]postulated a supercellfor NaCI whichcontained 16 conclusionholds even if the Debyetemperature is chosen particles,eight cations and eightanions. According to his

TABLE 5. Data for Optical Spectraof Simple Minerals and Framework Silicates

wt(0), wt(Kmax), Mineral cm-• cm-• Wu,cm -• we, cm-• q Comments

Halite 173 135 265 Periclase 390 303 730 Brucite 280 198 725 3670 0.13 Corundum 378 300 751 Spinel 225 178 600 700 0.22 model 1 225 178 750 model 2 a-Quartz 128 91 809 1100 0.22 a-Cristobalite see text 84 804 ! 100 0.22 Silicaglass see text 5 804 ! 100 0.22 model I (CRN) see text 14 805 1100 0.22 model2 (PD) Coesite 121 86 683 1100 0.22 model 1 121 86 792 1100 0.22 model 2 Stisho• itc 330 233 950 model 1 330 233 769 920 0.22 model 2 Rutile 113 80 840 model 1 113 80 500 610 0.05 model 2 824 0.17 Albite 83 63 648 1000 0.21 Microcline 74 62 648 1000 0.21 KIEFFER:THERMODYNAMICS AND LATTICE VIBRATIONS OF MINERALS, 3 47

(a) 1200 - (b) (c) (d)

e000-

800-

600- %.,..._

HALITE 400-

I i 201o I I I I

to:Li i I •oop•

600- 60 •......

PERICLASE 400- 40

200- o i I 0 I I i i i old/' i i i i i

5 //'""ßdehydration beginstOO0 0133 600

600 ,øøt ! o ' ' 1to.1

800

6O0

.CORUNDUM 400

200

I I I I i o

400

• • o.t9

6OO I1!MOOEL t% • /BOOEL2 400

200 4020.•t• •...... _L•' ...... - , , i I I 0 I I I I I O0 200 400 600 800 200 400 600 800 (ooo UO 200 400 600 800 1000 0 5 10 •5 20 25u(TM) I i I i C•'(cal. mol• (øK-i) vs.ll'q() Oc•l(øK)vs.T(øK) % contributionof bronches o •oo • • •,(• vs.T(øK) g(u)[orb. units] vs.u orw

Fig. 2. Data andmodel values for halite,periclase, brucite, corundum, and spinel. (a) Heatcapacities as a functionof temperature.(b) CalorimetricDeb.ye temperatures asa functionof temperature.(c) Percentcontribution to heatcapacity of acousticmodes A, opticcontinuum O, andEinstein oscillator E asa functionof temperature.(d) Modelspectra compared to latticedynamics calculations (for NaCl) or Raman(solid circles), infrared (open circles), or INS (squares)data. The verticalscale shown on spectra in Figures2, 3, and4 isarbitrary. Frequencies are given as t, (terahertz) and w (cm-•).The singularitiesin the acousticbranches at h, t,:,and ,8 areindicated by {hreedots on theacoustic branches. Parameters used in the modelare given in Tables4 and5. Referencesto heatcapacity data are given in Figure4 of paperI. Referencesto spectraldata are given in paper 2. 48 KIEFFER:THERMODYNAMICS ANDLATTICE VIBRATIONS OF MINERALS, 3

model, acousticmodes account for only 3/(3 X 16) = • of the 2c) it canbe seenthat thesedecreases are causedmainly by degreesof freedom, 'a small residue' of the total modes [Ra- anisotropyand dispersionin the acousticbranches. The sub- man, 1943, p. 250], in contrast to the 50% consistentwith the sequentsmall recovery of the O•(T) curvestoward 0•(0) is choiceof the Bravais cell as the fundamentalvibrating unit. due to the increasingcontribution of the optic modesas the Becausethe atoms in both halite and periclaseare rather temperature increases. homogeneously bonded, the wave velocities of these sub- The deviationof the periclasemodel heat capacityfrom stancesare nearly isotropic,although periclase exhibits consid- measured data above 200øK arises from the use of the K = 0 erably more anisotropy than halite. The shape of acoustic valueof 730 cm-• for the upperlimit of the optic continuum. branchesgiven by the assumedsine wave dispersionlaw agrees For this mode the K = 0 value is a maximum and the fre- well with the form of measureddispersion curves (see paper 2, quency tiecreasessubstantially (to •500 cm-• at the zone Figure 5) becausethe substancesare nearly'monatomic' in the boundaryin the [001] direction).Because the modesnear the senseof the criteria 1-4 in section3 of paper 1. zone centerare more heavily weightedin the calculationof the The optic modes of halite span a relatively small range of vibrational spectrum,the zone center value of the mode fre- frequencies(173-256 cm-• at K = 0). There are only three quencydoes not accuratelyrepresent the averagefrequency of optic modes(see Figure 2a of paper 2). The modesshow some the mode across the Brillouin zone. As can be seen from the dispersionthrough the Brillouin zone(particularly the longitu- actual vibrationalspectrum shown in Figure 2d, a value of dinal optic mode, which dips as low as 180 cm-• at K = 500-600 cm-• would be more representativeof the direction- 1.414(27r/a)in the [110]direction [Raunio et al., 1969,p. 1497]. ally averagedvalue of the LO mode; useof this value in the The simpledispersion relation adoptedfor the lowestTO optic modelgives heat capacitiesin excellentagreement with mea- branchin the model(w(Kmax) = w(0)/(1 + m2/m•)•/2, where sured values.Although many modesshow a similar decrease m2is the massof sodiumand m• the massof chlorine)predicts from a maximumvalue of frequencyat the zone centerto a a value of 135 cm-• for the TO branch at the zone edge, in lowervalue at the zone edge,some modes show the opposite reasonableagreement with the observedvalue of 142 cm-• (at behavior(see the dispersion relations in Figures2a-2e of paper K = 0.5(3•/:)(27r/a)in the [111] direction). 2). The degreeof dispersionand its trend dependon the The optic modesof periclaselie at higher frequenciesthan interatomicforces, and it is not possibleto specifya priori a those of halite becausethe Mg-O force constantsare greater simpledispersion curve for the high-frequencyoptic modes. than the Na-CI force constants and the atoms are less massive. To the extentthat the modesdecrease in frequencytoward the The LO and TO modesare relativelywidely spacedat K = 0: zone edge, the model, with vibrational frequencieschosen the LO mode is at 730 cm -• and the TO mode is at 390 cm -•. from opticaldata alone, will tend to give low estimatesof Cv As can be seenfrom measureddispersion curves (Figure 2b of (as occursin the caseof periclase)because the averagefre- paper 2) the zone center frequency of the LO branch is a quenciesof the optic brancheswill be somewhatlower than maximumfor the branch,and the frequencyfor the TO branch those specifiedin the model. A similar problem would be is a minimum for the branch. In the model the optic modes encounteredin the 'spectroscopicmodel' describedabove, in were representedby a continuumextending from 303 to 730 whicheach optic mode was represented by an Einsteinoscilla- cm-k the value 303 cm-• was obtainedby the correctionof the tor placedat the K = 0 frequencyof the mode; dispersionis K = 0 value of 390 cm-• for dispersionacross the Brillouin not accounted for in such a model. zone by the factor (1 + mo/ml)-•/•, where m• is the massof oxygenand m• is the massof magnesium. Brucite The model vibrational spectrafor thesetwo substancesare Brucite(Mg(OH)2) has one formula unit in the fundamental shown in Figure 2d, where they are comparedwith spectra Bravaisunit cell: it thereforehas 15 vibrational modes per unit obtained from detailed lattice vibrational calculations. The cell, three of which are acoustic. The addition of a formula vertical scalein thesefigures is arbitrary and is not the same unit of H:O to form brucite from periclaseresults in an in- for the variousspectra; therefore only positionsand relative creaseof the total number of degreesof freedomfor the unit magnitudesof featurescan be compared.Although the model cell from six to 15 and a reduction of the fraction of acoustic spectrahave an 'artificial' appearancearising from the sharp- modes from 50% to 20%. nessof the threeacoustic modes (which havesingularities at v•, The only acousticvelocity data available for brucite are and v•, and rs, shownschematically by dottedextensions of the shockwave data [Sirnakovet al., 1974]which give a sound modes)and from the arbitrarinessof the flat continuum,they velocity(interpreted to be a bulk soundspeed) of 4.75 km s-• are useful in showing the major features of the vibrational (correspondingto an elasticDebye temperature of 697øK).A spectra.They showgood qualitativeagreement with the more value of Poisson'sratio of 0.25 was usedto estimatevs and vp detailed spectraobtained from lattice dynamicscalculations: from this bulk soundspeed. Although the structureof brucite the extent of the acousticmodes is approximatelycorrect, as is highlyanisotropic, viz., cleavage,acoustic anisotropy could are the peaksin the spectradue to thesemodes; the overlapof not be reliably included in the model becauseof the uncer- acousticand optic modesat low frequencyis correctfor halite, tainty in the acoustic data. The acousticbranches calculated and the relative proportionsof acousticand optic modesare from thesewave velocitiespeak at 141 and 244 cm-• at the correct.The agreementis not asgood for periclasebecause the Brillouinzone boundary.Inelastic neutron scattering data modelspectrum includes too many oscillatorsat high frequen- showan acousticmode at 128cm -• [Saffordet al., 1963;Pelah cies (near the K = 0 longitudinal optic mode). et al., 1965]which may correspondto the top of the slow The heat capacityof halite is predictedwell by the model acoustic shear branch. spectrum;that of periclaseis predictedslightly less accurately. At K = 0 the opticmodes of brucite(280-725 cm -•) lie The 10ø drop in Oca•(T)at low temperaturefor halite and the relativelyclose to thoseof periclase(390-730 cm -•) exceptfor much larger decrease(80øK) in Oca•(T)for periclasematch the O-H stretchingmodes, which are at 3700 and 3655 cm-•. measured data. From examination of the relative contribu- Whenit iscorrected by thefactor (1 + m:/m•)-•/• (wherem• is tions of acousticand optic modesto the heatcapacity (Figure themass of magnesiumand m: isthe mass of oxygen),the zone KIEFFER:THERMODYNAMICS AND LATTICE VIBRATIONSOF MINERALS, 3 49 center frequencyof the lowestmode (280 cm-x) becomes215 acoustic modes are less dispersedthan the sinusoidal dis- cm-x at the zone boundary.The optic continuumwas there- persionlaw suggestsand, in particular, that shearmodes may fore taken from 215 to 725 cm-L The O-H modeswere ap- extend to frequencieshigher than the 184 and 198 cm-x pre- proximated as a single Einstein oscillator at 3670 cm-x con- dicted by such a law. A rough estimate suggeststhat these taining 13.3% of the total modes. The model vibrational modesmay be near 230 cm-x. spectrumis shown in Figure 2d. The heatcapacity of bruciteto 500øK is shownin Figure 2a. Spinel Above 500øK, dehydrationbegins, and the measuredheat Normal spinel(MgAI:O4) is cubic with two formula units capacity beginsincreasing rapidly owing to the onset of the (14 atoms) in the unit cell. The oxygen atoms are approxi- dehydrationreaction [Freund and Sperling, 1972].,(It should be mately in cubic closepacking. The magnesiumatoms are in notedhere that Co data given by Freund and Sperlingare positionsof fourfold coordination within a tetrahedral group approximately30% lower than the data of Giauqueand Archi- of oxygenatoms; the aluminum atoms are in sixfold coordina- bald [1937] usedin the figure.No explanationcan be givenfor tion within an octahedral group [Bragg et al., 1965, p. 103]. the discrepancy.)The heat capacityis predictedwell by the There are 42 degreesof freedom per unit cell, 3 of which (7%) model at low temperatures.The low-temperaturedecrease in are acoustic. Oca,(T) arises from the acoustic shear and longitudinal Like halite and periclase, spinel exhibits relatively little branches.Optic modesdo not contributeappreciably to Cv acousticanistropy becauseof its cubic symmetry.The shear until temperaturesreach -• 100øK. The O-H stretchingmodes velocities used are 5.31 and 6.00 km s-X; the compressional do not contributeto Cv at temperaturesbelow those at which velocity is 9.82 km s-x. The mean velocity is 6.24 km s-x, and dehydrationbegins. the acousticDebye temperature0• is 879øK. The optic modesextend from 225 cm-x (Raman active) to Corundum 750 cm-x (Raman and infrared active); a weak mode at 825 The Bravais unit cell of corundum (AI•.Os) contains two cm-x reported by Fraas and Moore [1972] is ignored here formula units of Alo.O8 and thereforehas 30 degreesof free- becauseit is probably related to sampleinpurities. It is inter- dom, 10%of whichbelong to acousticmodes. Anisotropy was estingthat although the highest-frequencybands of spinelare included in the acousticvelocities for the shearbranches, but it at exactlythe samewave numbersas thoseof corundum,they is minor. are not attributed to AI-O deformationsbut to Mg-O vibra- Numerous papershave beenpublished on the opticalvibra- tions [Preudhommeand Tarte, 1971a,b, c, 1972]. Thesevibra- tional modes of corundum. Well-documented infrared-active tions are reasonablywell isolatedfrom the rest of the bands modes exist at 635, 583, 569, 442, 400, and 385 cm-x, and well- which lie below 600 cm-x. Accordingly,they are modeledby documentedRaman-active modes at 751, 645, 578, 451,432, an Einsteinoscillator at 700 cm-x containing19% of the total 418, and 378 cm-x (seeTable 1 of paper 2). Inelasticneutron modes(see paper 2, Table 6). For comparisona secondmodel scattering(INS) data do not exist. It is of interest that the is shown in which thesemodes are includedas part of the optic modesof corundumdo not extendto nearlyas low frequencies continuum. The lowest observed mode at 225 cm-x is lowered as the modesof spinel(MgAlo. O4), perhapsan indication that to 178 cm-x at the Brillouin zone boundary by the assumed the lowest-frequencymodes in spinel are not simply AI-O dispersionrelation. The model vibrational spectrumfor spinel deformational modes. For the model it was assumed that the is shown in Figure 2d, where it is compared with IR and zone center frequencyof the lowestmode in the optic contin- Raman data. It should be noted that the spectrumis signifi- uum was 378 cm-x, which was reduced to 300 cm-x at the zone cantly different from the periclaseand corundum spectra at boundary by correction for dispersionacross the Brillouin low frequenciesand thus cannot be modeled as the simple zone (with rnataken as the massof oxygenand mx as the mass 'sum' of the.•etwo spectra.(This raisesthe interestingquestion of aluminum). The calculatedspectrum for the model is shown of whethercalorimetric properties of spinelcan be calculated in Figure 2d, where it is comparedwith IR and Raman data. as the simple sum of the component oxides, periclase and The heat capacity of corundum (shown in Figure 2a) is corundum.) The model spectrumalso showsclearly that in nearly Debyelike. There is a small decreasein O½a•(T)at 50ø- contrastto the simpler substancessuch as periclasethe acous- 100øK caused primarily by dispersionin the acoustic shear tic modescomprise only a small fraction of the total modesfor branches. The shear branches contribute to Cv in excessof the spinel. Debye behavior predicted from a mean sound speed. The The heat capacity Cv* and 0½a•(T)predicted by the model excessis offset,however, by relativelyhigh frequenciesfor the for spinel are given in Figures 2a and 2b. The propertiesare longitudinalacoustic branch and for the optic continuum.By similar to those of corundum except at low temperatures 300øK, O½•(T)rises back to a value only slightlylower than 0e•; where 0½a•(T)approaches a much lower value of 0•. The heat hencethe room temperaturespecific heat would be well pre- capacityis overestimatedby the modelin the low-temperature dicted by a 0e•based on elasticconstants even though Cv at range.Below 80øK the main contributionto the heat capacity room temperature is actually determined primarily by the is from the acoustic modes, and therefore the excess heat optic modes(see the relative contributionsin Figure 2c). The capacitypredicted by the model is probably due to an over- high-temperaturevalue of Ocli(T) of •900øK is characteristic estimateof the amount of dispersionin thesemodes. The low- of minerals with AI-O stretching modes as the highest fre- frequencyoptic modes contribute significantlyto the heat quency modes. capacity above 80øK, and the excessheat capacity of the A comparison of the model heat capacitieswith measured model in this rangemust thereforebe attributed to an excessof values near 50øK suggeststhat dispersion in the acoustic oscillatorsin the low-frequencypart of the optic continuum. branchesis overestimatedby the assumedsinusoidal dis- At high temperaturethe heat capacity of spinel is slightly persioncurves (equation (11)). This is an exampleof how the larger than that of corundum. This is most easily seen by heat capacitymodel developedhere may actuallybe usedto comparisonof the 0½a•(T)curves: 0•a•(øø) is •960 øK for corun- suggestfeatures of the vibrational spectrum,namely, that the dum and •875øK for spinel. This might initially be unex- 50 KIEFFER:THERMODYNAMICS ANDLATTICE VIBRATIONS OFMINERALS, 3 pected becausethe highest-frequencymodes for both sub- made up of extended three-dimensionalnetworks which lack stances are at 750 cm-'. A comparison of the model periodicity at any scale. Modern CRN theories (e.g., Evans vibrationalspectra in Figure2d revealsthe causeof thisbehav- and King, 1966; Bell and Dean, 1966] are specialcases of the ior: the optic continuumof spinel extendsto lower wave Bernal theory of the structure of liquids, a theory for the numbers(178 cm-' at the zoneboundary) than that of corun- random packing of monatomic spheres.CRN theoriesrequire dum (300 cm-' at the zone boundary),and becauseof this ex- that the structurebe essentiallyirregular throughout the crys- tensionof the optic continuumto low frequencies,there are tal. The randomnessof structurein a silica glassis attributed relativelyfewer oscillatorsat high frequenciesfor spinel.It to randomnessin the Si-O-Si angles and in the rotation of was demonstratedin paper 1 that not only is the frequency adjacent tetrahedra about the connectingSi-O bonds. CRN of the highest-frequencymodes important in determining theory holds that there is a definite difference between the the high-temperaturebehavior, but also the relativepropor- vitreous state and the crystalline state, not a gradational tion of the total modes at high frequency.In this case a changein structuralproperties between the two states. smaller proportion of the modesare at high frequencyfor The original microcrystallinetheory for glassstructure pre- spinel than for corundum. datesthe ZacharaisenCRN theory by severalyears [Randall et al., 1930]. The original suggestionwas that individual unit cells Quartz, Cristobalite, Glass,and Coesite of cristobalite(having dimensions on the orderof 7,4) were These SiO: polymorphswith fourfold coordinationprovide present in glass. The first systematicX ray investigationsof a test of the sensitivityof the model to structural variations glasses[Warren, 1933, 1934, 1937; Warren and Biscoe, 1938] and an opportunity to examinethe extensionof the model to demonstratedthat the presenceof suchcells was incompatible amorphoussubstances. The most importantstructural param- with X ray data. Hence the original microcrystallinetheory eters which vary are the unit cell volumesand the number of was discardedin favor of the then newly publishedZacharai- atoms in the unit cell. Variations in the acoustic velocities and sen CRN theory. in the degreeof anisotropycause minor variationsin the heat The modern microcrystallineview, which will be called the capacity. paracrysta!!inetheory here, maintainsthat in glasses,individ- Structure of SiO: polymorphs. The structuresof the crys- ual molecules,ions, or atomsare assembledinto small or large talline silicapolymorphs are well known [e.g., Sosman,1965] groups which have a high degreeof short-rangeorder com- and will not be reviewedhere. The crystallinephases--quartz, pared to the long-rangeorder in the glass[Valenkov and Porai- cristobalite,and coesite--have27, 36, and 72 degreesof free- Koshitz, 1936]. The structuralgroups can be regardedas small dom, respectively,per unit cell. The three acoustic modes distortedcrystals, separated from one another by zonesof less thereforecomprise 11% of the total modesfor quartz,8.3% for order. In contrastto the CRN theory the paracrystallinethe- cristobalite, and 4.2% for coesite. ory maintainsthat there is not a sharp distinctionbetween the The detailed structure of silica glass, however, is not well glassyand crystalline states. knownand hasbeen the subjectof muchcontroversy [Gaskell, For purposesof the thermodynamiccalculations I examined 1970;Bell et al., 1968;Bell andDean, 1970;Wong and Whalley, the consequencesof the paracrystalline model of Robinson 1970]. Except at very low temperaturesand frequencies,the [1965]. In this model it is proposedthat vitreousmaterials are heat capacityand Raman and infrared spectraare sufficiently made up of a jumble of n-sidedpolyhedra, each face of which similar to those same propertiesof quartz and cristobaliteto is a polygon with m sides.Glass is thereby consideredto be a suggestthat certain aspectsof lattice vibrational theory as grained, textured polycrystallinestructure composedof tiny developedfor crystallinematerials apply also to the vitreous cylinderswhose packing fraction is very high. By a seriesof state. Vibrational models based on the choice of the SiO4-4 arguments, Robinson concludedthat the polygonsare either tetrahedron as the fundamental vibrating unit have been suc- pentagons or hexagons, that is, m = 5 or 6. He chose to cessful in accounting for those features of the vibrational examine the structurecomposed of puckered pentagonaldo- spectrumwhich occur abovewave numbersof severalhundred decahedrawhich have 12 faces.The pentagonaldodecahedra cm-' [Gaskell, 1970; Bell et al., 1968; Bell and Dean, 1970]. havean equivalentradius of 3.91,4 andhave five SiO: mole- The work of Bell et al. [1968] and Bell and Dean [1970] and culesper dodecahedron.Randomness in a glasscomposed of the experiments of Leadbetter [1969], however, suggestthat pentagonaldodecahedra results from a randomnessin the O- although it may be theoretically possible to describe the Si-O anglesabout the tetrahedral value, some randomnessin higher-frequencyvibrations in glassesby consideringstruc- the Si-O-Si angle, and a randomness in the amount of dis- tural units similar to the crystallineanalogues, it is unlikely tortion in the individual polydodecahedra.Submicroscopic that the behavior of the low-frequencybranches will be simi- structural units of glass, such as those observedin transmis- larly modeled. sion electron microscope(TEM) bright field and dark field The two theoriesof glassstructure which satisfymost exist- micrographs[Prebus and Michener, 1954; Zarzycki and Me- ing data (X ray, transmissionelectron microscope,spectral, zard, 1962; Gaskell and Howie, 1974], were hypothesizedby and thermodynamic)differ substantiallyin their descriptionof Robinson to be tiny cylinderscontaining about 225 poly- long-rangeorder in glasses.These theoriesare referred to as dodecahedra(1125 SiO: molecules)each. Glass was thus be- the 'continuous random network' (CRN) theory and the lieved by Robinson to be comprisedof a grained, textured 'paracrystalline'or 'microcrystalline'theory. In both theories structureof thesecylinders whose packing fraction is very the basicstructural unit is acceptedto be the XY4 tetrahedron high.The densityof the submicroscopicstructural units (cylin- (e.g., SiO•-•), and adjacenttetrahedra are linked by common ders) is 2.29 g cm-3, and the 'microporosity'arising from Y atoms into a three-dimensional network. The nature of the imperfectcylindrical packing of the grainsis about 5%. bridgingbonds between tetrahedra is the subjectof the con- Neither the CRN nor the paracrystallinemodel can be troversy. treatedrigorously by a latticedynamics model becauseneither The continuousrandom network theory, first developedby modeladvocates the existenceof a unit cellregularly repeated Zacharaisen[1932, 1935]proposes that vitreousmaterials are throughoutthe glassby translationalperiodicity. The para- KIEFFER: THERMODYNAMICSAND LATTICE VIBRATIONS OF MINERALS, 3 51

(a) (b) (c) (d)

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/ t 4•// 8• 80• 3 60 60

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•1•'/ •• ...... 40•o• •• •1 •o• __L...... o•_'' ' ' ' ' o•_'' ' ' ' .' ol ...... • ' • • ' I I' •

•1•0, , , , ' •ool ' ' i i , oF- , , •• •o • 4o • •o...... /oo o • •o •o /oo a /o 2o o /o

* t ...... 00• 2• 400 600 8• ••0• ol0 •,, • •o •o •o ,o• o ...... 0 Cv* (cal. mole{ •-• vs.T(øK) •al(øK) vs.T(OK) % contributionvs.T•K)of broncos g(u)[a•.units] Fig. 3. Data and model values for quartz, cristobalite, glass, and coesite. (a) Heat capacitiesas a function of temperature.(b) CalorimetricDebye temperatures as a functionof temperature.(c) Percentcontribution to heatcapacity of acousticmodes A, optic continuumO, and Einsteinoscillator E as a functionof temperature.(d) Model spectracompared to optical data or INS data. The notation is the sameas that in Figure 2. crystallinemodel, however,may be looselyinterpreted to spec- dom per cylinder, three of which are acoustic.Although this ify a structurerather like a unit cell, e.g., the submicroscopic 0.03% is a very small fraction of the total modes,the contribu- cylindrical units of Robinson, consistingof 1125 SiO2 mole- tion of thesemodes would be measurableat low temperatures, culesin a •:ylinderof 25 • diameterand 100 • length.For this as is shown in Figure 3. choice of 'pseudo unit cell,' with 1125 SiO2 moleculesper The CRN modeldoes not lend itselfto treatmentby analogy grain, there are 9975 (approximately 10,000) degreesof free- to the lattice dynamics approach for crystals becauseof its 52 KIEFFER:THERMODYNAMICS AND LATTICE VIBRATIONSOF MINERALS, 3 basicprinciple that there is no small repeat unit in amorphous model for the separationof acousticand optic modesto the substances.In essense, in thismodel the structural unit of glass calculatedacoustic branches; this cutoff is loweredto 86 cm-• isthe macroscopic grain;in a typicalheat capacity experiment by the diSperSion assumed: across the Brillouin zone. The lowe r [e.g,, Flubacheret al., 1959]the macroscopicgrains are several cutoffis evenless well defined 'for cristobaliteand glass, al- millimetersin diameter.A descriptionof the lattice vibrations thoughin bothcases, Raman data or INS dataare available of such a substancecan be obtained only by solvingthe dy- (see paper 2). Far-infrared spectra obtained in this study namical matrix for the motions of a very large number of (paper 2) do not reveal any distinctabsorption bands below atoms (approximatelyAvogadro's number), thosein the mac- 298 cm-• for cristobalite or any bands below 470 cm-• for roscopicgrain. Current practiceis to make physicalmodels of glass(paper 2). Raman spectraof vitreoussilica [Flubacheret smallparts of the CRN structuresand to study the modesof al., 1959] show an intensecontinuum extendingcontinuously vibration of large numbersof atoms in thesephysical models from 560 down to 8 cm-•. This continuum was originally [e.g., Evans and King, 1966; Bell and Dean, 1966; Bell et al., attributed by Flubacheret al. [1959] to optical modesof very 1968]. The three acousticmodes belonging to such particles low frequency.Later inelasticneutron scattering work [Lead- are a negligiblefraction (< 10-24) of the total degreesof free- better, 1969] attributed the vibrational modesof glassat w •< dom and for all practical purposesdo not appear in a vibra- 40 cm -• to much broadened transverse acoustic branches. tional spectrumand do not contributemeasurably to the heat Becausethe neutron scattering differential cross section for capacity.To approximatethe CRN model, I have examineda fused silica was observed to be almost identical to that for model for glass in which the acousticmodes are l0 -6 of the cristobalite for w > 50 cm-•, it was suggestedthat the optic total modes;this fraction is picked arbitrarily (as one which mode distribution may be the same as that of cristobalite the computerprogram handleseasily). The readershould keep above this wave number. The neutron scatteringresults for in mind that the purposeof examiningthe heatcapacity of the both substancessuggest that the optic modesmerge with and silicapolymorphs, including glass, in the contextof the model are indistinguishablefrom the acousticmodes. It will therefore proposedfor heat capacitiesin this paper, is to examinethe be assumedin this model that the optic continuum of both consequencesof structural variations, particularly variations substances extends down to the lowest acoustic branch at the in the size of the unit cell. The treatmentof glassas proposed Brillouin zone boundary,that is, wt = w•. For cristobalitethis above shows the behavior of the model as the unit cell ap- meansthat wt is at 85 cm- •' for the CRN modelglass, wt is at 5 proachesinfinite size. cm-•, and for the paracrystallinemodel glass,wt = 13.5cm-x. Additional model data for the SiO2 polymorphs. The SiO• Results. The heat capacitiescalculated from the model are polymorphsare characterizedby relativelylow wave velocities shown in Figure 3a. The calculated heat capacitiesfor the becauseof their open structure and generally low densities. crystallinepolymorphs are in excellentagreement with mea- Shear velocitiesrange from 3 to 5 km s-•, and compressional sured values. In the case of coesite this must be considered velocities vary from 5 to 7 km s-•. Directionally averaged fortuitous in view of the assumptionsmade about the optic maximum and minimum shear velocities can be estimated for continuum. The only significantdeviations occur at temper- quartz from single-crystalelastic data [McSkimmin et al., aturesbelow 50øK. Even at very low temperatures(<20øK), 1965].The slow shearvelocity u• is estimatedto be 3.76 km s-•, where the acoustic branches make the dominant contribution and the fast shearvelocity u• is 4.46 km s-• (seethe appendixof to the heat capacityand where the thermodynamicproperties paper l). Slow and fast shearvelocities and Voigt-Reuss-Hill are sensitiveto detailsof the spectrum,the model resultsare in (VRH) shear and longitudinal velocities for all of the poly- good agreementwith measureddata for quartz and coesite. morphsare given in Table 4. The slow and fast shearvelocities The major discrepancybetween the model and the data is in given for glass,cristobalite, and coesiteare estimates(see the the region of temperatureswhere the lowest-frequencyoptic appendix of paper 1). modesmake an appreciableto major contributionto the heat The optic modes of quartz, cristobalite, and glass are re- capacity (20ø-100øK). For coesite the discrepancyis negli- markably similar at high frequencies.Si-O stretchingmodes gible; for quartz and cristobalitethe heat capacityis under- comprise22% of the total degreesof freedomin eachcase and estimated,as is shown best by the Oca•(T)curves (Figure 3b). occur in an isolated band centered near 1100 cm-•. There is a Examinationof the relative contributionsof acousticand optic weak absorption band in the infrared spectrumat 792 cm-• branchesshows that the model discrepancyarises from both which might correspondto the top of the optic continuumof branches:for the acousticbranches, anisotropy may be over- coesite(Figure la of paper 2). However, the heat capacity estimated, or, more likely dispersionis underestimated(see predictedby the model with this band as the top of the optic the discussionof spectra which follows), and for the optic continuum is a poor fit to measured data, and it will be branches,wt may be overestimated.At temperaturesabove suggestedbelow that the top of the optic continuumis at 683 100øK the optic continuum makes the major contribution to cm-•, the next highest observedmode in infrared spectra. In the heat capacity. The contribution of the Si-O stretching this casethe 794-cm-• band might be an impurity band (seethe modesis appreciableat room temperature. discussionof impurities in coesitein paper 2) or might be one The excellentagreement of the model heat capacitieswith of the stretchingmodes at anomalouslylow frequency;results measuredheat capacitiesdoes not of courseimply a similarly from both casesare given in Figure 3. detailedmatch of the model vibrational spectrawith measured The lower cutoff of the optic continuumis well definedonly vibrationalspectra. Nevertheless, some of the major important for quartz (oot= 128 cm-•). The infrared spectrumof coesite spectralfeatures are includedin the model,as may be seenby showsa continuousseries of optically active lines extending comparingthe measuredand calculatedspectra shown in the down to 265 cm -•. Raman data are not available. Because the figures.Consider the spectrumof quartz, for whichthe neut- two shearand one longitudinalacoustic branches extend to 85, ron scatteringexperiments of Elcorobe[1967, p. 954] have 107,and 167cm-•, respectively,it is highlyprobable that optic provideddispersion curves. The acousticvelocities used in the modesextend lower than the infrared data suggest.The value model and the assumedsine wave dispersiongive model acous- o0t= 121cm-• was estimatedby applyingthe one-dimensional tic brancheswhich are in reasonableagreement with the mea- KIEFFER:THERMODYNAMICS AND LATTICE VIBRATIONSOF MINERALS, 3 53 sured acousticdispersion curves found by Elcombe[1967, p. silica and of Leadbetter and Litchinsky [1970] on vitreous 954]. The frequenciesat the zone boundaryand the assumed germania, have led to a different interpretation--that the ex- sinusoidalshapes are qualitativelycorrect, but on the average cesslow-frequency mode is a broadenedtransverse acoustic the model appearsto underestimatesomewhat the amount of branch. dispersion.Elcombe's experimental data for the normalmodes In the model of this paper the low-frequency modes are in the z direction and Leadbetter's[1969] determinationof the treated formally as optic modesbelonging to the optic contin- spatially averaged dispersioncurve place the lowest shear uum. This continuum extends down to and is continuous with branch in the range 55-80 cm-•, somewhatbelow the model the acoustic branches;however, the density of state changes value of 103 cm -•. The same trends are noted for cristobalite. abruptlyat the optic continuumbecause for both the continu- Inelasticneutron scatteringexperiments on cristobalite[Lead- ousrandom network (CRN) and the pentagonaldodecahedral better, 1969]indicate truncation of the shearbranches at 30-50 model(PD) the acousticmodes are a very small fraction of the cm-• and 60-85 cm-•, substantiallybelow the model valuesof total modes(• 10-6 in the computermodel for the CRN and 85 and 101 cm-•, respectively.Leadbetter comments that the •1(, -8 in the PD model). From Figure 3b it can be seenthat lowest shear branch is highly dispersedand flattens off at the PD model with its smaller 'unit cell' than the CRN 'cell' remarkably low energy. more closelyapproximates the heat capacity;a unit cell con- Both models discussedfor glass predict a heat capacity taining a few hundred atoms, rather than a few thousand, which is far greater than that of quartz or cristobalite at would give the correct behavior at low temperatures. For temperaturesof a few degreesKelvin (seeFigure 3b), although comparison,a calculationwith the 'supercell'of a-tridymite both modelsslightly overestimate the amount of excess.This containing 72 atoms is shown in Figure 3b. This unit cell, excessheat capacity is well known and appearsto be charac- which is the largestof the crystallinesilica polymorphs,is too teristic of amorphous materials in general [e.g., Anderson, small to account for the excessheat capacity. 1959; Fritzsche, 1973, p. l l8; Antoniou and Morrison, 1965; Even thoughthe model in no way representsrigorously the Robie et al., 1978]. The measuredexcess heat capacity may be physicsof glassvibrations, it is stronglysuggestive that the expressedas the sum of two terms with different behavior: a excesslow-temperature heat capacitycould be explainedsim- term linear with temperatureand a term cubic with temper- ply on the basisof low-frequencyoptic modes at frequenciesin ature and of larger magnitude than is expectedfrom simple the spectrumwhere they are expectedfrom simple structural Debye theory [Fritzsche,1973]: considerations,i.e., from the general occurrenceof low-frequency modes in substanceswhich have large unit cells. A Cv = AT + B7• (32) simple test of this idea would be comparisonof low-temper- The specificheat proportional to T• arises from a phonon ature heat capacitiesof glassesof varying thermal histories densityof statesg(w) proportional to w:; however,the coeffi- (and therefore,presumably, varying structures)with high-res- cient of proportionality is not that obtained from the elastic olution TEM micrographs.The excessheat capacityshould be constants.The linear term dominatesbelow about 1øK [Zeller related to the size of any paracrystallitesobserved. and Pohl, 1971];the coefficientA is 10 ergsg-• øK: to within a Rutlie and Stishovite factor of 2. The linear term can be derived from a density of statesspectrum of phononexcitations which is independent of R utile(TiO•.) and stishovite(SiOn.) have primitive tetragonal energy(as is the optic continuumof this paper) with a density structureswith six atoms per primitive unit cell. The titanium of the order of 10:1 cm -8 eV -1 between 0 < E < 2 X 10 -8 eV and silicon atoms are the centers of octahedra of six oxygen [Fritzsche,1973, p. 119]. The causeof both excessheat capac- atoms which occur at approximatelyequal distancesfrom the ity termshas beenthe subjectof muchdebate in the literature. cations. The unit cell of rutile (64.42 X 10-•'4 cm8) is consid- Flubacheret al. [1959] and Leadbetter[1969] concludedthat erably larger than that of stishovite(46.54 X 10-•'4 cm8) be- excessphonon modes at low frequencieswere required to causetitanium is larger than silicon. Although the geometric explainthe low-temperatureheat capacityand postulatedthat arrangementof the rutile structuregives little indication of thesemodes arise from two sources:(1) localizedmodes at w < anisotropyin bonding,the elasticconstants are anisotropic.It 20 cm-1 and perhapsas low as 6-9 cm-1 arising from struc- has beensuggested in the literature that the structureof rutlie tural defectssuch as oxygenvacancies and (2) a mode at about belongsto a transitiontype betweenionic and moleculartypes 40 cm-1 containing about 1.5% of the total modes. This 40- [Seitz, 1940, p. 50; Matossi, 1951]. Thus descriptionsof the cm-i mode is also observedin cristobalite, in which it appears vibrations in terms of modified TiO•. molecules are not uncom- to contain 2-3% of the total modes.There is strong evidence mon, and there might be somejustification for searchingfor that at very low energies(w < 20 cm-1) localized modes 'intramolecular' Ti-O modes of vibration, as suggestedin pa- associatedwith structural defectssuch as oxygenvacancies are per 2. important.These modes comprise about 1 in 104of the degrees From detailed measurementson anisotropicwave velocities of freedom and account for the linear term in (32) [Leadbetter, in single-crystalrutile [Robieand Edwards,1966] it is possible 1969].The causeof the excessheat capacityproportional to Ta to obtain reasonablygood estimatesof the directionally aver- has not been so well established.Smythe et al. [1953] and agedshear wave velocities.The velocitieswere calculated(ap- Anderson [1959] proposed that low-frequency bending of a pendix of paper 1) to be u• = 4.73 and u•. = 5.73 km s-•. weak elongatedSi-O-Si bond contributedto the excessspecific Maximum and minimum shear velocities measured in high- heat. Andersonand Bi•mmel [1955] argued that this mode was symmetrydirections are /)mln-- 3.31 and /)max-- 6.76 km s-• the cause of the low-temperatureacoustic absorption in vit- [Manghnani,1969; Robie and Edwards, 1966]. The largediffer- reous silica. Clark and Strakna [1962] showed that the elon- ence between the averagedvelocities u• and u•. and the large gated Si-O-Si bond acts as a Schottky type of defect and can difference between the minimum and maximum velocities/)mln explain that part of the excessheat capacity below 30øK. and Vmax,in high-symmetrydirections, both show the large However, the neutron scattering experimentsof Wong and degreeof acousticanisotropy characteristic of rutile. Similarly Whalley [1970] and Leadbetter [1969] on various forms of detailed measurements are not available for stishovite, and it 54 KIEFFER:THERMODYNAMICS ANDLATTICE VIBRATIONS OF MINERALS, 3 •_ (a) ,•oj-(b) (c) 1ooor MO•Lt 1oo ...... o RUTILE•oo- •ot'\,..'

I ' ' ' s 200o-- ' ,oo-, • 800'•//.....--- 80 O

2 400- 40

t 200 20 MODEL t o o . ,?

0205 F

4•',, iI / 8oo/•/ 80 3 600 • ALBITE4oo • ' ,•o

,I-/! oo_ ,• ...... o1;, , ,_.: , o. , , , , , o' ......

0 2O5 5 i//// 1000- 4J.-'v. / ,• 800- '•"1 • / / •00-/ •0 • MICROCLINE4 4o

\ ...... • J J I • I / I I I I I O' • -'m""-----I------.,1- --- .•I------I O• 200 400 600 800 1000O0 200 400 600 800 1000 0 200 400 600 800 t000 t0 t5 20 25 30 u(Thz.) C•(col. mol• 1øK'I) vs.T(øK) • •o 8•o •000' w (cnf •)' vs.T(øK} g(u)[orb units] vs.u or w Fig. 4. Data and model values for rutile, stishovite,albite, and microcline.(a) Heat capacitiesas a function of temperature.(b) CalorimetricDebye temperature as a functionof temperature.(c) Percentcontribution to heatcapacity of acousticmodes A, opticcontinuum O, and Einsteinoscillator E asa functionof temperature.(d) Model spectracompared to optical data or INS data. The notationis the sameas that in Figure 2.

wastherefore necessary to estimate the acoustic anisotropy by This mode is one of three modes which are not active under comparisonwith rutile(appendix of paper1 ). The meansound Raman or infrared experimental conditions. Another low- speed/-)mand Debye temperature are higher for stishovite (6.16 frequencymode, the /12u(TO) mode, is of specialinterest km s-• and921øK) than for rutile(5.72 km s-• and781øK). becauseits frequencyis highly temperaturedependent. The Acousticbranches of rutilehave been measured by inelastic frequencyof thismode drops from 172cm -• at roomtemper- neutronscattering [Traylor et al., 1971](see Figure 2e of paper atureto 140cm -• at 4øK. Althoughthis extreme behavior may 2). Someof the branchesare approximately sinusoidal (e.g., be peculiar to rutlie becausethe c axis static dielectricconstant branchesin the[100] direction); others deviate markedly (e.g., is exceptionallyhigh [Trayloret al., 1971],it is not uncommon thelongitudinal branch in the [110]direction). Anisotropy is that the frequencyof modesdepends on temperature.Spectra verypronounced, for example,the differencein slopesof the obtained at room temperatureand assumedto hold at low longitudinalacoustic branches in the[100] and [001] is nearly temperaturemay not reproducelow-temperature heat capaci- a factor of 2. tiesbecause of changesin the low-frequencypart of the spec- Therutile INS dataalso show well the distribution of optic trum. modes,their dispersioncurves, and theirrelationship to in- The optic modesof rutile extendto 841 cm-• in a fairly fraredand Raman data. The lowestoptical mode is at l l3 continuousdistribution(see Figure 2 of paper2 or Figure4d cm-• at K = 0 and is relativelyflat acrossthe Brillouinzone. of thispaper). The threehighest modes have been assigned as KIEFFER:THERMODYNAMICS AND LATTICE VIBRATIONSOF MINERALS, 3 55

Ti-O stretchingmodes [Traylor et al., 1971], and it is likely tions would be at approximately50 cm-•, i.e., in optic modes that the mode at 610 cm-• is a stretchingmode (seediscussion or anomalies in the acoustic branches at about this wave in paper 2). The stretching modes thus probably span the number. Such modesdo not appear plausible because(1) the region 600-850 cm-•. This is a relatively broad frequency acoustic velocities of stishovite are so high that even the ex- range, and it is substantiallylower than the range 1000-1200 treme rutilelike anisotropy assumeddoes not give a low-fre- cm-• generally spanned by Si-O stretching vibrations, for quencyshear branch anomaly at 50 cm-•; (2) dispersiondoes which it was found that representationof the vibrations by a not generallyplay a role until w • 0.5w•, i.e., to about 80 cm-• singleEinstein oscillatorat, say, 1100 cm-• introducedrela- in this case;and (3) optic modesdo not generallylie below the tively little error into C•, calculationsbetween 0 ø and 1000øK. lowest shear branch which, in the case of stishovite, is at 158 For rutile, detailsof the spectrumat •700 cm-• influencethe cm-• at the Brillouin zone boundary. However, if the low- heat capacity at temperaturesas low as 300øK. For com- frequencymodes observed in rutile were characteristicof min- parison,two modelsare shown:model 1 with all optic modes erals with the rutile structure, a low-frequency optic mode in the optic continuumand model 2 with one mode(0.056 of below w• would be possible.Simple cation massscaling from the total modes)at 610 cm-• and three modes(0.167 of the the frequencyof the lowestrutile mode would suggestthat this total modes)at 824 cm-•. The two modelsshow appreciable mode would be near W/,rutileX (mq:•/msi)•/•' • 150 cm-1, far differences above 200øK. above the wave number of 50 cm-• required to explain the The spectrumused in the model is comparedin Figure 4 excessheat capacity. It thereforeseems unlikely that the drop with a spectrumobtained from the shellmodel of Trayloret al. of the measured0•l(T) to the low minimum is explicableby [1971]by fitting of INS data. Althoughat low frequenciesthe anisotropy, dispersion, or low-frequency optic modes. An- two modelsagree in the representationof the optic continuum, other cause for this anomaly was suggestedby Holm et al. the spectrumof this paperis significantlydifferent from that of [1967], who obtained the calorimetric data: that the excess the Traylor model in the positionof the acousticpeaks. The heat capacity arises from surface contributions becausethe Traylor modeldoes not reproducethe measuredelastic Debye meanparticle size in theheat capacity experiments was 740 • temperature:0e• from that modelis 660øK, whereasthe mea- [Holm et al., 1967].To the extentthat thesemodel calculations sured value is 781øK. The model used in this paper has the do not match the low-temperatureCv data when reasonable correct acousticlimit, and thus the acousticpeaks are at con- lattice behavior is assumed,the model considerationssuggest siderablyhigher frequencies than thoseof the Traylor model. that the excessheat capacitycould be from nonlattice effects The rangeof optic modesand their distributionare lesswell such as surface energy. The model may therefore give an known for stishovitethan for rutile becausespectral data are appreciablybetter estimateof the lattice heat capacitybelow incomplete.The lowestwell-documented infrared mode is at 50øK than the data. 330 cm-• (it would drop to 233 cm-• at the zone boundary 2. Above 40øK the model overestimatesthe heat capacity becauseof the assumeddispersion). Striefier and Barsch [ 1976] in comparison to calorimetric data. Because optic modes have suggesteda mode at 190 cm-• from rigid ion model dominate in this temperaturerange, the behavior must be due calculations; it will be demonstrated below that such a mode to an excessof optic modesat frequenciesbetween 200 and 300 would not be consistentwith the measuredheat capacitiesof cm-•; i.e., it is probably due to a poor estimate of the lower stishovite.The highestinfrared-active modes in stishoviteare cutoff frequency of the optic continuum. Use of the Striefier at 950 cm-•. As for rutile, two models were tried: model 1 with and Barsch[1976] value of 190 cm-• for wt would only increase all of the optic modesin a continuumextending to 950 cm-• the discrepancybetween the model valuesand the calorimetric and model 2 with the optic continuumtruncated at 769 cm-• value. and the Si-O stretchingmodes as 22% of the total modesat 920 In the case of stishovite there are uncertainties in all of the cm -•. data examined: acousticvelocities, spectral data, and calori- The heat capacityfor rutile is givenin Figure 4a; the agree- metric data. Therefore each data set should be subjected to ment with measureddata is good, and the two modelsexam- reexamination.Nevertheless, the model proposedgives an esti- ined bracketthe high-temperaturebehavior. From Figure4c it mate of the heat capacitywhich is probably more realisticthan can be seenthat at temperaturesabove 50øK the heat capacity the Debye model, givessome insight into the vibrational spec- is dominatedby the optic mode distribution.Rutile is unusual trum, and affords more detailed comparisonwith the proper- in that the lowest optic modes (at or near w = 113 cm-•) ties of rutile than does the Debye model. extend below the lowest averaged shear branch (w• = 158 A lbite and Microcline cm-•). These modescause the extremedrop in Oca•(T)at low temperatures.The minimum value of Oca•(T)is obtained at Albite (NaA1Si3Os) and microcline (KAISi3Os) contain 52 25ø-30øK; optic modescontribute approximately 30% of Cv atoms in their monoclinic unit cells. The state of A1-Si disorder at this temperature. affects the thermodynamic properties but is not considered The two models examined for stishovitediffer only in their here becausedetailed descriptionsot' the samplesused for the behavior above 300øK, and becauseno calorimetric data exist acousticand optical parametersare not available.An excellent at higher temperatures,a preferredmodel cannot be selected. discussionof the effect of the ordering parameter on ther- For convenience,only model 1 (with all optic modesin the modynamic properties is given by Openshaw[1974]. For all optic continuum) will be discussedfurther. At low temper- practical purposesthis paper assumescompletely ordered al- atures the model does not match measured calorimetric data. bite and microcline. 1. Below 40øK the model predicts a much lower heat Becausethere are 156 degreesof freedom in the unit cell, the capacitythan was measuredby Holm et al. [1967]. In this three acousticbranches account for only 2% of the total vibra- rangeof temperatures,only the acousticmodes contribute to tional modes. The acoustic velocities of the feldspars are Cv becausethe optic modesare at muchtoo high frequencyto among the lowest measured for minerals. The difference in contributesubstantially. (Even Strieflerand Barsch'smode at velocitiesbetween albite and microcline is insignificant in the 190 cm-• would not contribute to Cv below 40øK.) If this model. Becauseof the low acousticvelocities and large unit excessheat capacitywere due to latticevibrations, the vibra- cell (small Brillouin zone) the acousticbranches are limited to 56 KIEFFER:THERMODYNAMICS AND LATTICE VIBRATIONSOF MINERALS, 3 verylow wave numbers: 45, 60,and 92 cm-• for w•,w2, and wa orthosilicates;pressure and temperaturedependences of heat in albite and 45, 47, and 89 cm-• in microcline. capacity,entropy, enthalpy, and Helmholtz free energywill be As mightbe expected from these low values for the acoustic given in paper 5 of this series. peaksand from the discussion in paper 2, opticmodes occur at The fundamentalassumption underlying the theory devel- very low frequencies.A weak infrared mode occurs at 83 oped here is that the basic vibrating unit of a crystal is the cm-• in albite; a well-documented infrared mode occurs at 98 primitive unit cell. From this assumption,which is rooted in cm-• in microcline, and liishi et al. [1971] suggestedan even Bloch's theorem of translational periodicity, it follows that lower mode at 74 cm -•. This lower mode was used in the acousticmodes comprise 3 out of 3s degreesof freedom,where modelbecause although it was not documentedin the Kieffer s is the number of atoms in a unit cell. Because minerals observations(see Figure 1 of paper 2), the discussionbelow generally have large unit cellswith many atoms, the acoustic suggeststhat it may be a Raman-activemode observedfor modes are generally a minor fraction of the total modes. somereason by Iiishi et al. in the infrared experiment.These Therefore acousticdata alone reveallittle about the thermody- low modes are associated with K-O or N a-O cation-oxygen namic properties of complex substancesexcept at temper- vibrations[liishi et al., 1971]. The modesare loweredby the atures of a few degreesKelvin. assumeddispersion to 63 and 62 cm-• for albite and micro- The model spectraare only approximationsto real spectra cline, respectively.Optic modesarising from Si(Al)-O bend- but are correct in their enumeration of the fraction of the ing, deformation,and torsionextend out to •800 cm-• [liishi modeswhich are acousticand in representationof the actual et al., 1971,p. 108].These modes are representedby the optic regionsof frequencyin which there are vibrational modes.The continuum in the model. The 20.5% of the modes attributed spectraprovide good estimatesof the heat capacitybecause to Si(AI)-O stretching(in paper 2) cover a broad band from the thermodynamic functions are averagesover the whole 720 to 1050 cm-• in both minerals.They are representedby a frequencydistribution and are insensitiveto details of the singleEinstein oscillatorat 1000 cm-• in this paper. spectrum(except at low temperature).An approximatingspec- The heat capacitiespredicted by the model are shown in trum such as the one proposed,valid for a wide variety of Figure 4a, wherethey are comparedwith measureddata. For minerals,is a usefultool for correlatingthermodynamic prop- albite, optic modescontribute half of the heat capacityby erties with elastic properties, optical spectra, and inelastic 20øK; by this temperature,the heat capacityhas nearly risen neutron scatteringdata. back to the value of the elastic Debye temperature,471øK. Structure and composition of minerals influencethe ther- The subsequentrise of Oca•(T)between 30øK and 400ø-500øK modynamicproperties in the following ways. is causedby the spreadof bendingand deformationmodes 1. The densityof the mineral is generallydirectly related (a over a wide range of frequencies.The model heat capacityis monotonicallyincreasing function) to the acousticvelocities slightly too large at high temperatures,but on the whole the and therefore to the slopesof the acousticbranches. representationis muchbetter than a Debye model.(It might be 2. The volume of the unit cell determines the size of the of interest to the reader that when the lowest IR band of albite Brillouin zone and thuq the proportion and extent of the at 92 cm-• was initially usedin the model,the heatcapacity at acousticbranches. Large unit cellshave small Brillouin zones 20ø-50øK was not accurately reproduced. This led me to and thereforelow-frequency cutoffs for the acousticbranches. hypothesizethat the slight shoulder at 83 cm-• on the IR 3. The number of atoms in a unit cell determines the total spectrum (Figure ld of paper 2) was a Raman-active, IR- numberof degreesof freedomand thus the fractionswhich are forbidden line. J. Delany subsequentlyobtained a Raman in acousticand optic modes. spectrumon albite and found the 83-cm-• band.Although the 4. Atomic masses and force constants determine the abso- real strengthof the model lies in usingspectral data to calcu- lute valuesof lattice vibrational frequencies;atomic massand late the calorimetricfunctions, I believethat this capabilityof force constant ratios determine the 'stopping band' between usingthe calorimetricdata to infer limits on the spectraillus- acousticand optic modes. trates the very strong relationshipbetween the spectroscopic 5. Tightly bound clusterswithin silicatesmay contain an and the thermodynamic data.) appreciablefraction of the total modesat isolatedfrequencies. The importanceof the low-frequencyoptic vibrationsmust The fraction of vibrations associated with intramolecular be reemphasizedhere. The distributionof theseoptic vibra- stretchingmodes is easily calculablefrom simple analysisof tions, which arise from cation-oxygenmotions, dominates the the structure. heat capacitybehavior in the range 30ø-100øK. This is pre- The investigationof the mineral heat capacitiesin terms of ciselythe range of temperatureswhich contributesmost heav- the model has revealedthe followinggeneral sensitivities of the ily to the entropy. Therefore an accurateestimate of the en- heat capacitiesto spectraldetails: tropy, say, S298ø, cannot be made from acousticwave velocities 1. The specificheat below about 50øK cannot be repro- but only from a reasonablycomplete knowledge of the low- duced accurately unlessspectral details below 150 cm-• are frequency optic mode distribution. knownto a resolutionof about 10 cm-•; hencemodel accuracy may be limited to +50% becauseneither acousticnor optic spectraldata are known with sufficientprecision in this range 6. SUMMARY of wave numbers. A formfor a simplifiedvibrational spectrum generally appli- 2. The specificheat above 50øK can be predictedreason- cableto mineralshas been proposed and applied to a number ably well if the general range spanned by optic modes is of simpleminerals and frameworksilicates in thispaper. The known. The model accuracyis typically +5% at 298øK for purpose was to examine systematicrelationships between mineralsfor which IR spectraare available. structure, composition,and the thermodynamicfunctions. 3. The specificheat at high temperature,above 300øK, is The temperaturedependence of the heat capacitywas exam- relativelyinsensitive to detailsof the high-frequencypart of ined here and will be given in paper4 for chain,sheet, and the spectrum.In particular,it is generallyadequate to model KIEFFER: THERMODYNAMICSAND LATTICE VIBRATIONSOF MINERALS, 3 57 the Si-O stretchingmodes, which may cover a range of 300 gE frequency distribution for an Einstein oscilla- cm-•, by a singleEinstein oscillator at a mean frequency.The tor, (18). model typically has an accuracyof + 1% at 700øK. h Planck constant,equal to 6.625 X 10-:7 ergs s. Applicability of the model to a large number of minerals Planck constant divided by 2•r. and accuracyfor thoseto which it is applied is limited by the j,j' cell index, before (1). paucity of data at wave numbersbelow 200 cm-• and the k Boltzmann constant, equal to 1.380 X 10-x6 sensitivityof the thermodynamicproperties to the details of ergs deg- x. the vibrational spectrumat low frequencies.The lower cutoff K(n) wave vector, before (1) ((2) in paper 1). frequencyof the optic continuum has probably rarely been Kmax maximum wave vector (Brillouin zone bound- observedin optical spectra for two reasons:(1) there is a ary), (8). generallack of low-frequencyoptical data on minerals;such X (xX•)optic continuum function, (23). data, at K = 0, would providean estimateof wt; and (2) from mr particle mass,(1). theoreticalconsiderations the lowestvalues of wt are expected NA Avogadro's number, equal to 6.023 X 10•'• at K = Kmaxrather than at K = 0 (seeFigure 7 of paper 1 and mol-•, before (1). Figure 2 of paper 2). Therefore the accuracyto which low- n number of atoms in the chemical formula on temperaturethermodynamic properties (which dependheavily which the molar volume is defined, before (1). on the value of wt) can be predictedis limited if INS data are q proportion of vibrational modesin Einsteinos- not available and only data at K = 0 are available. cillator, (18). The model is usefulin the form presentedhere as a method R gasconstant per mole,equal to 1.988cal mol-• of predicting thermodynamic properties of minerals; the deg-X), (21). method is independentof any calorimetricdata. Becauseof r, r' base index, before (1). nonharmonic contributions to the thermodynamicproperties s number of particlesin Bravais unit cell, before not accountedfor in the model, the calculationscannot replace (•). calorimetricdeterminations, but will be primarily usefulin (1) S* entropy (molar) normalized to monatomic providinggood estimates of the thermodynamicproperties of equivalent, (27). materialsfor which there are insufficientsamples, for samples $(xt) dispersedacoustic function, (22). which are of poor quality for calorimetricwork, or for materi- T temperature, degrees Kelvin unless otherwise als which are unstable, e.g., stishovite,and (2) providing a indicated, (7). framework to interrelate thermodynamic, spectroscopic, ut spatiallyaveraged acoustic velocity for acoustic acoustic, and structural data. branchesi, where i = 1, 2, 3, before (10). v(O,½) acousticvelocity, (10). v NOTATION molar volume, (8). volume of unit cell, (8). vR volume of unit cell of reciprocal lattice (Bril- Equation numbersgiven refer to the first useof the symbol 1ouin zone), (8). in an equation or to the equationsnearest the first occurrence w wave number, equal to w/2•rc cm-•, between of the symbol in the text. (25) and (26). Ill, a9., aa basicvectors of the primitive lattice, before(1). Wl, W2 ,W3 wave numbers of acoustic branches at Brillouin A coefficientfor low-temperatureCv of glass,(32) zone boundary, where i = l, 2 for shear only. branchesand i = 3 for longitudinal branch. B bulk modulus, (29). W.(r) wave amplitude, (4). B coefficientfor low-temperatureCv of glass,(32) x(/) general position vector, before (1). only. x(/) distanceof unit cell from origin, before (1). c speedof light. x(r) position of a particle within a unit cell, before lattice potential coefficients,(5). (•). C•, heat capacity(molar) at constantpressure, (29). dimensionlessfrequency (t•/kT), (22). heat capacity(molar) at constantvolume (7). reciprocallattice vector, equal to K/2, after CvE heat capacity(molar)of an Einstein oscillator (4). (same as •;(14w/kT)in paper 1 and as •;(xE)), z number of 'molecules' in Bravais unit cell, be- (24). fore (1). Cv* heat capacity (molar) normalized to mon- summation index, (1•-(6) only. atomic equivalent, (21). summationindex, (2)-(6) only. E* internal energy (molar) normalized to mon- thermal Gri•neisenparameter, (31). atomic equivalent,(26). Kronecker delta function, (6). Einstein function, (7)and (24). - Dirac delta fi•nction, (18). Helmholtz free energy (molar) normalized to thermal expansion, (29). monatomic equivalent, (28). Oca,(T) calorimetricDebye temperature,after (6). g(K), g(w) frequency distribution function, between (6) 0o Debye temperature,after (31). and (7). elasticDebye temperature,after (31). g,(K), gt(w) frequency distribution of acoustic branch i, (0, ½) directionangles, (10). where i = 1, 2, 3, (12a). •'mln cutoff wavelength.(9). go frequency distribution of optical continuum, displacementof a pointat x in the crystal;com- (19). ponents/•.(/), a = 1, 2, 3, (1). 58 KIEFFER: THERMODYNAMICSAND LATTICE VIBRATIONSOF MINERALS, 3

v frequency,cycles per secondor terahertz, be- heat of SiOn.glass, Phys. Chem. Glasses,3(4), 121-126, 1962. tween (4) and (5). Clark, S. P., Handbook of PhysicalConstants, Mem. 97, Geological Society of America, Boulder,Colo., 1966. Po, Pr densityat 0øK and at temperatureT, (31). Cochran, W., Lattice dynamicsof ionic and covalentcrystals, Crit. potentialenergy of the lattice,(1). Rev. Solid State Sci., 2(1), 1-44, 1971. angularfrequency, radians per second,(4). De Sorbo, W., The effect of lattice anisotropy on low-temperature COE Einsteinfrequency, (18). specificheat, Acta Met., 2, 274-283, 1954. •(K) Dixon, A. E., A.D. B. Woods, and B. N. Brockhouse, Frequency Frequencyof branchi, wherei = 1, 2, ... , 3s, distribution of the lattice vibrations in sodium, Proc. Phys. Soc. after(6); fromsection 3 to endof paper,refers London, 81, 973-974, 1963. to acousticbranches, i = 1, 2, 3. Elcombe,M. H., Someaspects of the lattice dynamicsof quartz, Proc. cot,max maximumfrequency of acousticbranch i, i = 1, Phys. Soc. London, 91, 947-958, 1967. 2, 3, (11); after (17) shortenedto cot. Evans, D. L., and S. V. King, Random network model of vitreous silica, Nature London, 212, 1353-1354, 1966. cot lower frequencylimit of optical continuum, Flubacher, P., A. J. Leadbetter, J. A. Morrison, and B. P. Stoicheft, (19). The low-temperatureheat capacity and the Raman and Brillouin cou upper frequencylimit of optical continuum, spectraof vitreoussilica, J. Phys. Chem. Solids,12, 53-65, 1959. (19). Fraas, L. M., and J. E. Moore, Raman selection rule violation for a • solid angle, (10). spinel crystal,Rev. Brasil. Fis., 2(3), 299-310, 1972. Freund, F., and V. Sperling,Hydrogen bondingin Mg(OH)•. and its effecton specificheat and thermal expansion,in ThermalAnalyses ProceedingsThird ICTA, Davos, Switzerland, edited by H. G. Wiedemann, vol. 2, pp. 381-392, Birkhauser, Basel, Switzerland, Acknowledgments. In the early days of this work (nearly 10 years 1972. ago), Barclay Kamb gave generouslyof his time, not as one man, Fritzsche, H., A review of some electronicproperties of amorphous but as all six of Kipling's honest servingmen: his name was 'What substances,in Electron and Structural Properties of Amorphous and Why and When, and How and Where and Who.' I gratefully Semiconductors,edited by P. G. Le Comber and J. Mort, pp. 55- acknowledgethose many fun hours of relentlessquestioning and 123, Academic, New York, 1973. spirited discussion.This work was begun as a Ph.D. dissertationat Gaskell, P. H., Vibrational spectraof simple silicateglasses, Discuss. the California Institute of Technology (CIT); the manuscriptswere Faraday Soc., 50, 82-93, 1970. completed while the author was an Alfred P. Sloan Research Fellow Gaskell, P. H., and A. Howie, High resolutionelectron microscopy of at UCLA. I thank the Sloan Foundation for support which allowed amorphoussemiconductors and oxideglasses, in Proceedingsof the the time and freedom for culmination of the project. The efforts of 12thInternational Conference on Physicsof Semiconductors,July 15- Art Boettcher as AssociateEditor in securingmany reviews of this 19, 1974, editedby M. H. Pilkuhn,pp. 1076-1080,Truebner, Stutt- paper and the helpful and thought-provoking comments and ques- gart, West Germany, 1974. tions by Roger Burns and other reviewerswere greatly appreciated, Giauque, W. F., and R. C. Archibald,The entropyof water from the although the author bears sole responsibilityfor errors which may third law of thermodynamics:The dissociationpressure and calori- remain. Barbara Niles (CIT), B. Gola, Gen Kurtin, Barbara Nielsen, metric heat of the reaction Mg(OH)•. = MgO = H•.O: The heat and Vicki Doyle-Jones(UCLA) contributedextraordinary efforts in capacitiesof Mg(OH)•. and MgO from 20 to 300øK,J. Amer. 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