earth thermal conductivity

PHYSICS OF THE SOLID EARTH� English Translation� VOL� �� NO� �� FEBRUARY �

Russian Edition� JULY AUGUST �

Temperature dependence of lattice thermal

conductivity of Earth�s mineral substance

G� I� Petrunin and V� G� Popov

Moscow University

Abstract� The speci�c features of the temperature dependence of lattice thermalconductivity of rocks and minerals in the range from room temperature to theirmelting points have been studied and generalized on the basis of experimentaldata obtained in the geothermal laboratory of the Physical Department� MoscowUniversity� Discussion in the context of thermal conductivity theory for a crystallinedielectric allows us to relate such features to the complexity of compositional andstructural imperfections of real mineral substances� Estimates can then be made ofthe in�uence of these factors on phonon scattering and mean free path� The analysisconducted here allows basic principles for constructing pro�les of conductive heattransfer in the crust and mantle to be developed�

In considering the experimental results of studies ofheat transfer in rocks and minerals� the principal fea�ture is the thermal conduction theory for crystallinedielectrics� which was formulated by Debye and devel�oped by Peierls and other investigators on the basis ofquantum mechanical principles� It is quite clear thatthe majority of geological materials representing poly�crystalline aggregates should be well suited to the ap�plication of theoretical conclusions on the character oftemperature dependence of thermal conductivity thatare based on the idea that the heat transfer intensitydue to lattice oscillations is governed by the degree ofanharmonicity of these oscillations� However� it is im�portant to bear in mind that the theory developed fora perfect crystal fails to agree quantitatively with ex�perimental results even for single crystals with a smallconcentration of structural defects and will yield stillmore discrepancies with data for complex mineral com�pounds that are �imperfect� in every respect� Thus�to extend the theoretical conclusions to real materialsand� in particular� to crust and mantle materials� thetheory should be supplemented by characteristics re�vealed through comparison with experiment� The spe�ci�c properties of materials are exhibited� as a rule� inanalyses of the change in thermal properties through awide temperature interval� which includes a high tem�perature range that has been only poorly studied untilrecently� The chief cause of this situation is� brie�y�the great diculty of running high�temperature exper�iments at increasing temperatures and with increasing

Copyright �� by the American Geophysical Union�

��

emission loss from the surfaces of the samples understudy�

Leibfried and Schlamann derived a well�known ex�pression �which relates temperature and other proper�ties of material� to describe the phonon thermal conduc�tivity of an ideal dielectric� For three�phonon acousticscattering at T � �� this formula can be written in theform MacPherson and Schloessin� ��

�p ��

��

�k

h

�M��

��T� C

M��

��T��

where � is the Gruneisen parameter� � is the volume re�ferred to one atom� � is the characteristic temperature�and M is the atomic mass or� for a complex substance�the mean atomic mass� As mentioned above� this for�mula does not give a quantitative agreement with ex�periments� and therefore a norming constant C has beenintroduced to �t �� to experimental data for each struc�tural type of compound�

Formula �� appears to provide qualitative theoreticalsupport for Eucken�s empirical law �� which showsthat the thermal resistivity ��l� of a crystalline di�electric at T � � is directly proportional to its absolutetemperature

��l � const T ��

Here� we consider experimental data on the thermalproperties of minerals and rocks and use our recentresults together with data from the literature Birchand Clark� �� Clauser� �� Fujisawa et al�� Kanamori et al�� Kawada� �� MacPherson and

Schloessin� �� Magnitskiy et al�� Petrunin and

��

�� petrunin and popov� temperature dependence

Figure �� Temperature dependence of the thermal re�sistivity of the major rock�forming minerals� �� plagio�clase �labradorite�� microcline� �� garnet� �� pyroxen�ite� �� synthesized gallious garnet� �� garnet Kanamoriet al�� synthesized olivine Schatz and Sim�mons� �� olivine �� Kanamori et al�� SiO� Kanamori et al�� and �� olivine Fo�� P � �� GPa Scharmeli� ��

Popov� �� Petrunin et al�� Scharmeli� ��Schatz and Simmons� �� Shown in Figures � and �are the temperature dependencies of the heat resistivi�ties of the major rock�forming minerals and magmaticrocks� respectively� First� it is beyond question thatover practically all the represented temperature inter�vals� the heat resistivity of the complex crystalline com�pounds forming the crust and mantle increases by a lawthat is also close to linear� however� this law di�ers from�� and has the following functional form�

�� A� BT ��

This relation involves� along with the second term de�scribing thermal resistivity due to the intrinsic phonon�phonon scattering in a crystal� an additional tempera�ture�independent term A that varies over wide limits fordi�erent minerals and rocks� It is easily seen that ex�cept for quartz� at room temperature this term is com�parable to or even exceeds the thermal resistivity con�tribution related to the anharmonicity of elastic latticeoscillations� Naturally� the �rst term in �� cannot be

Figure �� Temperature dependence of the thermal re�sistivity of some rocks� �� granite� �� granite Birchand Clark� �� diorite� �� diabase� �� dunite�� dunite� �� gabbro Birch and Clark� �� serpenti�nite� �� hypersthenite Birch and Clark� �� eclog�ite Kawada� �� and �� mica Kanamori et al��

ignored in estimating the thermal conductivity withinthe Earth�s mantle�

In the lower right�hand corner of Figure �� we com�pare �� and �� using them to extrapolate the �� Kdata for dunite to the higher temperatures characteristicof the deep upper mantle� hence it follows that neglectof the A term results in signi�cant overestimation �bytwo times� of the thermal resistivity� This was one of thereasons that in the ��s� geophysicists concluded thatthe Earth�s mantle transfers heat conductively with rel�atively low eciency� Thus consideration of the e�ect ofthis term on the predicted thermal conductivity of Earthmaterials requires careful study of their physical natureand estimation of their contribution to the total ther�mal resistivity at di�erent horizons of the mantle� Asfor the physical nature of A� it is natural to propose itsrelation to imperfections in the materials and primarilyto scattering of phonons at microdefects such as grainboundaries� cracks� voids� etc� Previously Petrunin and

Popov� �� we made an e�ort to justify this assump�tion by demonstrating the in�uence of the homogeneityof a material on the value of thermal resistivity� includ�

petrunin and popov� temperature dependence ��

ing the additional temperature�independent term� Itwas shown that for particular minerals and rocks� thisin�uence is substantially suppressed on passing from�ne�grained to coarse�grained structures and� especially�to monocrystals and monomineral aggregates� In thesame work we discussed experimental Fujisawa et al��

�� results on simultaneous pressure and temperaturee�ects on polycrystalline forsterite and concluded thatthe thermal resistivity decrease obtained by these au�thors is mainly related to enhancing grain boundarycontacts and likely cannot be induced by pressure e�ecton the phonon�phonon interactions� since at pressuresP � � and � GPa� the slope of the thermal resistiv�ity versus temperature curve remains constant withinthe limits of experimental errors� A similar conclusioncan be reached from an analysis of isobaric dependencies��T � constructed from the results ofMacPherson and

Schloessin �� who demonstrated the P�T e�ects onthe thermal conductivity of synthesized monocrystals ofpericlase �MgO� and galite �NaCl� at temperatures of�� K and pressures of �� to �� GPa� However�in the case under consideration� the weak decrease of the�defective� part of the thermal resistivity A is accom�panied by a faintly visible tendency for the slope of thetangent to the isobars to decrease with increasing pres�sure� Considering the relation of the slope determinedby the coecient B in �� with structural character�istics such as the lattice constant� Debye temperature�and Gruneisen parameter� we can suggest that the pres�sure range used is not suciently large to appreciablychange these parameters� Thus at present� the ques�tion of pressure variation in the normalizing constant inB and consequently in the coecient B itself remainsopen because of insucient experimental data� Thislack casts some doubt on the possibility that relations�� and �� are applicable to predicting the thermal con�ductivity of mantle materials�

We here continue our discussion of the physical natureof the coecient A� Once the e�ect of macrodefects onthe value of this coecient has been established� a validquestion about the degree of this in�uence arises� Thisin�uence can be estimated from comparing the temper�ature dependencies of poly� and monocrystals of sub�stances� However� in this case� we are interested in thepurely qualitative matter of the mechanism of the e�ectrather than in numerical estimates for various minerals�In other words� can the value of A be completely ex�plained by the macrodefect e�ect� or is part of it inducedby some other phonon�scattering mechanism Such aquestion arises because� starting from the temperaturebehavior of monocrystalline olivines and garnets repre�sented in Figure �� we can assert that phonon scatteringat macrodefects does not seem to exhaust the value ofthe A term and is substantially caused by other scatter�ing mechanisms�

Figure �� Temperature dependence of heat capacityof rocks� �� granite� �� diorite� �� diabase� �� gabbro�� dunite� �� harzburgite� and �� serpentinite�

First and foremost we dwell on the cause that hasnothing to do with thermal resistivity� According tothe theory of an ideal dielectric� the law ��l � T ap�plies in a temperature range above the characteristictemperature T � �� where the heat capacity reachesits classic limit and the temperature behavior of thelattice thermal conductivity is exceptionally well de�termined by transport processes with anharmonic cou�pling� The value of � for the considered minerals androcks lies within the limits �� K� consequently� inthe studied temperature interval �� K�� and es�pecially within its �rst half� heat capacity continues togrow� resulting in a decrease in the slope ��l � f�T �and an increase in A� This e�ect must be greater forhigher characteristic temperatures �� i�e�� for greater in�creases in the heat capacity� It follows from Figure ��which shows heat capacity as a function of temperaturefor ultrabasic rocks that are similar in composition tothe upper mantle� that the heat capacity over the indi�cated temperature interval increases almost �� times�and the proposed e�ect must be signi�cant� This e�ectcan be estimated using measurements of thermal dif�

fusivity a ��

�v!l� which is directly proportional to the

mean free path !l of phonons and must well character�ize the phonon scattering processes and the associatedconstraint on thermal conductivity� By analyzing datafrom the literature as well as our direct measurements ofthe thermal di�usivity for di�erent minerals and rocks�Figures � and �� it is easily shown that the contribu�tion of the values of A� to ��a at �� K is smaller bynearly two times than the similar contribution of A to��l �Figures � and �� for the same materials� For ex�ample� at �� K� these contributions for the olivine andsynthesized garnet monocrystals amount to �� and ��"�Figure �� respectively� in contrast to the correspond�ing �� and ��" contributions in the case of thermalresistivity �Figure �� A similar result for rocks can be


Figure �� Temperature dependence of the quantity ��afor some rock�forming minerals� �� plagioclase� �� mi�crocline� �� olivine Kanamori et al�� olivine�� jadeite Kanamori et al�� pyroxenite� �� gar�net Kanamori et al�� quartz Kanamori et al�� forsterite Schatz and Simmons� �� and�� gallious synthesized garnet�

obtained by comparing the corresponding dependenciesdepicted in Figures � and �� This implies that the pro�cesses of phonon scattering by structural imperfectionsare responsible for only� ��" of the value of the A termin �� Moreover� the fact that ��a is nonvanishing atT � � for the garnet and olivine single crystals� whichare close to ideal and practically macrodefect�free sam�ples� suggests that apart from macrodefects� the �defec�tive� part of thermal resistivity is contributed not onlyby the structural imperfections of foreign atom or va�cancy types but also by phonon scattering from localchanges in the density and elastic modulus characteris�tics of chemical compounds� In particular� this hypoth�esis is not in con�ict with the data in Figures � and�� From these data for �� and ��a as well as for Aand A�� we recognize a certain tendency� namely� thesevalues increase with increasingly complicating chemicalcomposition of the crystal lattice� from close to zerofor SiO� through an intermediate value for pyroxenesand olivines �MgSiO�� Mg�SiO�� to the maximal val�ues which almost completely determine the thermal re�sistivity of complex solid solutions of plagioclases andfeldspars� This hypothesis is important from the pointof view of both theory and practice� but it needs spe�cial investigation and closer experimental veri�cation�This investigation can also shed light on the unusual

Figure �� Temperature dependence of ��a for rocks�� granite� �� diorite� �� diabase� �� gabbro� �� dunite�� harzburgite� �� serpentine� �� basalt� �� pyroxenite�� enstatite� �� peridotite� and �� eclogite�

temperature behavior of heat conductance and thermaldi�usivity of some minerals and rocks� which at �rstseems contrary to the fundamentals of lattice heat trans�fer in a lattice dielectric� This speci�c feature can bemore distinctly traced at high temperatures and con�sists of the characteristic �attening of the correspond�ing curves at temperatures of �� K and� in somecases� the tendency of � and a to increase with increas�ing temperature� This is especially clear in Figures �� and � when we consider deviations of the high�temperature points from linear dependence �� f�T �and ��a � f�T � over the measured temperature interval�� K�

We can argue with con�dence that such behavior ofthe e�ective thermal conductivity and thermal di�usiv�ity of minerals and rocks in the high�temperature rangeis not speci�c fact but experimental regularity� On theother hand� it is dicult to imagine that there is onlyone general cause for this behavior� This is connected tothe facts that data from di�erent authors for the samematerial often disagree and that the trends of the de�viations of heat resistivity from linear dependence varyin character� In particular� for olivine and garnet� thebehavior of dependencies �� f�T � and ��a � f�T ��Figures � and �� according to Kanamori et al� ��essentially di�ers from the behavior seen in our labora�


tory� For practically all minerals measured by Kanamoriet al�� the low temperature of �� K is the thresholdabove which we see a substantial departure from lineardependence and a subsequent sharp increase in thermalconductivity and thermal di�usivity� We are inclinedto consider this fact as being due merely to methodrather than a methodical e�ect being unrelated to ma�terial properties� and later on we shall try to unravelthis situation in more detail�

In our opinion� there are at least three principalcauses for the phenomenon studied� each of which candominate depending on method of study� applied theproperties of the substance� and structure� composition�and thermodynamic conditions of the experiment� Thepossibility that two or even all three causes will operatesimultaneously is also not inconceivable�

The �rst cause is the way temperature increases dueto heat radiating from the sample surface a�ect the re�sults of measurements� This is a pure methodologicalcause related especially to the diculties of account�ing for the indicated heat losses� It can be shown thatfor materials with low conductivity� including the rock�forming minerals� in a given experiment with a speci��ed object geometry� the dimensionless Biots parameterBi characterizing the intensity of the radiative heat ex�change can reach a tenth at temperatures of �� K�This leads to signi�cant overestimation of thermal con�ductivity and thermal di�usivity up to reversing thetrend in the temperature behavior� Contrary to thetechnique ofKanamori et al� �� our technique takesinto consideration this heat transfer and excludes its in��uence on measurement results Petrunin and Jurchak�

�� This suggests that it might be realistic to explainthe deviation �observed even at moderate temperatures�of the mineral thermal di�usivity curves published byKanamori et al� �� methodological�

The second cause is an appearance� along with the lat�tice thermal conductance� of a new heat transfer mech�anism� Such a mechanism in a crystalline lattice and�in particular� in mineral and rock can be the heat radi�ation mechanism with intensity proportional to T � andactivated in transparent or half�transparent bodies fromthe beginning of their luminescence �� K�� Ra�diative heat transfer in mineral substances is a specialproblem of thermal physics that has been clearly formu�lated and continues to be developed in many laborato�ries� including geophysical ones Clauser� �� Fujisawaet al�� Kanamori et al�� MacPherson and

Schloessin� �� Scharmeli� �� and others�� Here wenote only that the principal criterion for identi�cation ofthis mechanism� in our opinion� is a stable tendency forthe measured values of thermal di�usivity to increasewith increasing temperature� It is the growth of ther�mal di�usivity �not thermal conductivity� that results inthe e�ect� The problem is that because of anharmonic

vibrations of atoms about lattice points� the isobaricheat capacity Cp continues to grow also at temperatureshigher than the characteristic temperature� inducing amonotonic increase in the e�ective thermal conductiv�

ity �ef ��

�Cp�v!l even if the radiation mechanism is

absent and the free path of phonons !l remains constantfor unknown reasons�

This may explain why the mean free path of phononsreaches its minimal value !l � vmin� In reality� themean lifetime of phonons is determined by scatter�ing processes� among which the three�phonon Umklappprocesses become dominant� According to the theory�the frequency of such processes increases proportion�ally with temperature and shortens the mean phononlifetime� The minimal value min is determined by themaximal frequency of atomic vibrations� which in a realdiscrete lattice structure must in turn be bounded bythe Debye spectrum cut�o� frequency�

max �k

h� ��

Thus for the minimal value of min� we have� by de�ni�tion�

min ��

max

�h

k��

and� consequently�

lmin � vh�k� ��

The thermal Debye temperature � can be replaced�with a sucient accuracy� by its acoustic approxima�tion�

� �h

k

��Na�

��M

��

v ��

where Na is Avogadro�s number� � is density� and v isthe mean sound velocity de�ned by the relation betweenthe longitudinal vp and transverse vs components�

v � � �v�pv

�s

v�s � �v�p��

Table � gives the results of computations of lmin formajor minerals and rocks in the Earth and comparableresults obtained from experimental thermal di�usivitydata obtained by using the formula

a ��

�v!l ��

It is easily seen that such a comparison leads to an un�expected result� even at room temperature� the meanfree path of phonons in the majority of geologic materi�als �except quartz� proves to be only slightly in excess


Table �� Mean free path of phonons in minerals and rocks as computed from thermal di�usivity data

Formula� Mean v ��Mineral� rock composition atomic km#s kg#m�� !l��$A !l��$A !l��$A lmin�$A

mass M

Olivine� �� Scharmeli �� Fo��Fa� �� Forsterite� MacPherson and Mg�SiO� �� Schloessin ��Olivine Fo��Fa� �� Quartz� Scharmeli �� SiO� �� Jadeite� Scharmeli �� NaAlSi�O �� Garnet� Scharmeli �� Pyroxene� Scharmeli �� En�Fs� �� Microcline� Petrunin et al� �� KAlSi�O �� Labradorite� Petrunin et al� �� Ab�An� �� Dunite Ol�� Harzburgite Ol�En� �� Eclogite Gr�En� �� Diorite � �� Gabbro � ��

Parentheses indicate the average values for a given type of rocks and minerals according to Maj �� and Povarennykh et al�

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of the minimal value that is reached in the temperaturerange �� K�

This fact fundamentally changes our present view ofhigh�temperature behavior of lattice thermal conductiv�ity and thermal di�usivity of such compounds as com�plex in their structure and composition as minerals androcks� First� it sets a limiting temperature of the orderof �� K for application to rocks and minerals notonly for the Leibfried and Schlemann formula obtainedfor T � � but for the more general empirical relation�� as well�

The same fact allowed us to begin developing the prin�ciples of numerical estimation of conductive heat trans�fer in the deep mantle on the basis of the known relation

� � �Cpa ��

��Cpvlmin ��

where lmin is determined from �� and the in�uence ofthermodynamic conditions is taken into account by pa�rameters � and v� obtained from relevant seismic mod�els� It follows from �� that with allowance for all of thepreceding� the conductive heat transfer in the mantlemust continuously increase �corresponding to the deepbehavior of v and �� rather than decrease with depth�

Thermal conductivity pro�les computed with the helpof �� for the simpli�ed PEM seismic models of theupper and lower mantle Dziewonski et al�� aregiven by Fujisawa et al� �� and Petrunin and Orlik

��

References

Birch� F�� and H� Clark� The thermal conductivity of rocksand its dependence upon temperature and composition�Am� J� Sci�� P� �� and � P� ��

Clauser� C�� Opacity The concept of radiative thermal con�ductivity� in Handbook of Terrestrial Heat�Flow DensityDetermination� pp� �� Kluwer Academic Publish�ers� ��

Dziewonski� A� M�� A� L� Hales� and E� R� Lapwood� Para�metrically simple Earth model consistent with geophysicaldata� Phys� Earth Planet� Inter��

Fujisawa� H�� N� Fujii� H� Mizutani� H� Kanamori� andS� Akimoto� Thermal di�usivity of Mg�SiO�� Fe�SiO��and NaCl at high pressures and temperatures� J� Geophys�Res��

Jurchak� R� P�� et al�� Study of Thermophysical Propertiesof Dielectrics at High Temperatures� in Teplozicheskiyesvoystva tverdykh veshchestv Thermophysical Propertiesof Solids�� pp� �� Nauka� Moscow� ��

Kanamori� H�� N� Fujii� and H� Mizutani� Thermal di�usivitymeasurement of rock�forming minerals from �� to �� K�J� Geophys� Res��

Kawada� K�� Studies of thermal state of the Earth� The �thpaper� Variation of thermal conductivity of rocks� Part ��Bull� Earth Res� Inst� Univ� Tokyo��

MacPherson� W� R�� and H� H� Schloessin� Apparent� latticeand radiative thermal conductivity at temperatures from�� to �� K and pressures up to �� GPa� Results for


MgO and NaCl� in High Temperatures�High Pressures� ��pp� ��

Magnitskiy� V� A�� G� I� Petrunin� and R� P� Jurchak� Behav�ior of the thermal di�usivity of some �eld spars and pla�gioclases at temperatures �� K� Doklady AN SSSR� ��

Maj� S�� Przewodnicto ciepla i temperatura w gornym pla�szczu ziemi� Acta Geophysica Polonica� ��

Petrunin� G� I�� and R� P� Jurchak� Allowance for e�ect ofthe radiative heat exchange in thermal di�usivity mea�surements by the plane temperature waves method� Vest�nik MGU Ser� � Fiz� Astron��

Petrunin� G� I�� and E� V� Orlik� Conductive heat transfer inthe Earth�s mantle� Doklady AN SSSR� ��

Petrunin� G� I�� and V� G� Popov� On phonon thermal con�ductivity at the upper mantle interfaces� Izv� AN SSSR�Fizika Zemli� ��

Petrunin� G� I�� V� G� Popov� and M� I� Timoshechkin� Tem�perature behavior of heat capacity� thermal di�usivity andthermal conductivity of gallium garnets� TVT� � � ��

Povarennykh� A� S�� G� T� Prodayvoda� and A� Ju� Serga�On the importance of the averaged elastic constants ofminerals� Mineral� Sh� L�vov��

Scharmeli� G� H�� Anisotropy of olivine thermal conductivityof �� GPa and up to �� K measured on optically non�thick samples� in High�Pressure Research in Geoscience�edited by W� Schreyer� pp� �� Stuttgart� Germany��

Schatz� J� F�� and G� Simmons� Thermal conductivity ofEarth materials at high temperatures� J� Geophys� Res��

�Received June ��

earth thermal conductivity

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