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, spectro- scopic,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 fundamental vibrating 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