The lattice thermoconductivity of an ideal crystal is obtained as a function of temperature by using a radiation-like description of the heat carriers (Debye phonons).Opisujući nosioce topline (Debyeve fonone) analogno ponašanju fotona dobivena je temperaturna ovisnost toplinske vodljivosti rešetke u idealnom kristalu

Apostol, Marian

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ISSN1330–0008CODENFIZAE4LETTER TO THE EDITORON THE LATTICE THERMOCONDUCTIVITY OF AN IDEAL CRYSTALMARIAN APOSTOL1Centre de Recherches sur les Très Basses Températures, CNRS, BP 166 X.,38042 Grenoble-Cédex 9, FranceReceived 14 September 1992UDC 538.931Original scientific paperThe lattice thermoconductivity of an ideal crystal is obtained as a function of tem-perature by using a radiation-like description of the heat carriers (Debye phonons).It is well-known from classical arguments1) that the lattice thermoconductiv-ity of an ideal crystal as a function of the temperature T behaves like T 3 atlow temperatures (boundary scattering limit)2) and obeys a 1/T -law in the high-temperature limit3) with an in-between peak which is usually referred to as theumklapp-scattering contribution4). This behaviour is obtained in the present paperby computing the lattice thermoconductivity of an ideal crystal in the framework ofthe radiation-like description of the Debye phonons. In addition, the thermoconduc-tivity corresponding to the non-equilibrium transport of energy is obtained, whichbehaves like 1/T 2 in the high-temperature limit. The radiation like description ofthe phonons is based on the natural assumption that the carriers are absorbed,during the energy transfer process, at a rate which is proportional to their thermaloccupation number. The standpoint expressed by Peierls4) is taken as regards theideal crystal, in that the phonons are viewed as well-defined elementary excitations.The effect of various collision processes is shown how to be included in the presentapproach.1On leave from the Department of Theoretical Physics, Institute of Atomic Physics, Magurele-Bucharest MG-6, PO Box MG-35, Romania.FIZIKA A 1 (1992) 2, 175–180 175apostol: on the lattice thermoconductivity . . .As in the usual experiments of measuring the thermoconductivity, consider arod of an ideal crystal, oriented along the z-axis and having a length much greaterthan its transverse size d. Under these circumstances one assumes that the thermalequilibrium is attained. A slight temperature difference ∆T is overimposed alongthe rod, such that the heat flows only along the z-axis. The ~q-th mode in a phononbranch with the frequency ω(~q) contributes (∂ω(~q)n(~q))/∂T )∆T to the density ofthe flowing heat, where n(~q) is the Planck–Einstein distribution. This energy iscarried along the z-axis with the group velocityvz(~q) = ∂ω(~q)/∂qz; (1)therefore, the average heat flux along this axis isQ(~q) =∂∂T[ω(~q)n(~q)]· vz(~q)τz(~q) ·∆T , (2)where τz(~q) is the corresponding relaxation time. The total flow of heat (heatcurrent) is obtained from the continuity equation and by summing over all thephonon modes,∂Q∂t=1(2π)3∂∂z{∫d~q∂∂T[ω(~q)n(~q)]v2z(~q)τz(~q) ·∆T}, (3)whence one obtains the thermoconductivityK =1(2π)3∫d~q∂∂T[ω(~q)n(~q)]· v2z(~q)τz(~q) . (4)In the case of a uniform transport vz(~q) = v2/3, τz(~q) = τ , one obtains the classicalresult K = CqΛ/3, where C is the specific heat, v2 is the mean square velocity andΛ = vτ is the relaxation length (or mean free path). For a classical gas of particlesΛ is the average inter-particle spacing and, as such, it is a constant as long as thenumber of particles is fixed. This is no longer true in the case of lattice vibrationswhere the number of phonons is not fixed and the heat transfer mechanism isdifferent.In the low-temperature limit the energy transfer is controlled by the collisionsagainst the boundary walls. In this case it is well-known2) that the (non–thermal)relaxation time is given byσ0z(~q) =dv⊥(~q), (5)where v⊥(~q) is the group velocity along the transverse directions and d is of theorder of the transverse size of the rod (shape-effects included). At non-vanishingtemperatures the absorption probability is increased by the n(~q) factor, according176 FIZIKA A 1 (1992) 2, 175–180apostol: on the lattice thermoconductivity . . .to the radiation-like description of the phonons. Therefore, the (thermal) relaxationtime is reduced toτ thz (~q) =dv⊥(~q)n(~q). (6)The usual Matthiessen rule5)1τz(~q)=1τ0z (~q)+1τ thz (~q), (7)may be used now, which interpolates between the thermal and non-thermal regime,with the resultτz(~q) =dv⊥(~q)[1 + n(~q)]−1. (8)The thermoconductivity (4) can now be written asK =d(2π)3T 2∫d~qω2(~q)exp[ω(~q)/T]− 1·vz(~q)v⊥(~q), (9)which, for a Debye spectrum to ω(~q) = vq, becomesK =9π4dnvQ(ωD/T ) ,Q(x) =1x3x∫0dξξ4eξ − 1,(10)where ωD = (6π2n)1/3v is the Debye frequency and n is the number of unit cells perunit volume (a factor of 3 polarizations has also been included). The asymptoticbehaviours of the thermoconductivity (10) are readily obtained asKT≪ωD∼= 54πdnv(TωD)3≃ 0.73Cdv ,KT≫ωD∼=9π16dnvωDT.(11)where C is the specific heat. The reduced thermoconductivity Q given by (10) isplotted versus T/ωD in Fig. 1, where one can see the peak located at T.= 0.47ωD.It is noteworthy that the high-temperature 1/T–behaviour in (11) arises alsofrom the low-frequency phonons, which control the relaxation length in this high-temperature region by their excessive thermal activation. Therefore, one may saythat not only the T 3–law in the low-temperature limit but also the 1/T–law inthe high-temperature limit is a consequence of the quantum nature of the heatFIZIKA A 1 (1992) 2, 175–180 177apostol: on the lattice thermoconductivity . . .Fig. 1. The reduced thermoconductivity Q given by Eq. (10) versus the reducedtemperature T/ωD.carriers. One may also add that for an anisotropic Debye spectrum ω(~q) = (v2‖q2z +v2⊥q2⊥)1/2 the factor v in (10) is replaced by the ratio v2‖/v⊥. In general, Eq. (9) canbe used for any phonon branch, anisotropy included6).In the case of non-equilibrium transport of energy (but preserving the localthermal equilibrium) one can derive, in the same manner that led to Eq. (4), thethermoconductivityK̄ =d(2π)3∂∂T∫d~q ω(~q) ·v2z(~q)v⊥(~q)n(~q)1 + n(~q), (12)related to the energy flow. It coincides with K in the low-temperature limit butbehaves likeK̄ =9π20dnv(ωD/T )2 (13)in the high-temperature limit, where the energy transport is limited by the dissi-pation processes.In conclusion, one may say that the lattice thermoconductivity of a rod of anideal crystal has been derived without making any particular assumption on thespecific mechanisms of heat transfer, except that the phonons are absorbed at therod boundaries at a rate which is proportional to their occupation number. This isthe natural assumption as long as the radiation-like description of the ideal phononsholds. However, various heat transfer mechanisms can be included in this picture,their effect being that of further reducing the relaxation time given by Eq. (8). Forexample, one may expect such a reduction to arise from the electron-phonon in-teraction. However, the number of electron states available for interaction is much178 FIZIKA A 1 (1992) 2, 175–180apostol: on the lattice thermoconductivity . . .smaller than the number of phonon states, the usual estimation being the ratioof the sound velocity to the Fermi velocity. Therefore, one may say that the lat-tice thermoconductivity is not much affected by the electron-phonon interaction.The presence of the impurities may also be included in Eq. (8). In particular,the well-known infrared divergencies1) coming from the scattering length of theimpurities are, according to (8), screened out, such that the thermoconductivitypeak will be more or less affected, but not the asymptotic behaviours. The effectof the phonon-phonon interaction and of the crystal anharmonicities may also betaken into account. As long as these processes involve higher frequencies (as forthe umklapp-scattering) they will mainly affect the peak of the thermoconductiv-ity, decreasing the magnitude and shifting its position toward lower temperatures.Only where the low-frequency part of the Debye spectrum is damaged one mayexpect a departure from the ideal asymptotic laws (11). All these effects can mostconveniently be accounted for by means of various techniques7−11), among whichthe well-known variational approach to the Boltzmann equation. An interestingpoint would be to check Eqs. (10) and (11) against the available experimentaldata. This can be done most conveniently for compounds with low Debye fre-quency ωD. Among the quasi-one-dimensional compounds one can often find thebest candidates. A very interesting problem occurs, however, in this case, regardingthe phason thermoconductivity (and, generally, the thermoconductivity of a latticesoft-mode). Equation (4) suits very well the anisotropy of these compounds, suchthat the computations might successfully be carried out. All these matters are leftfor forthcoming investigations.AcknowledgementsThe author thanks A. Smontara who prompted him to undertake this investi-gation, and J. C. Lasjaunias and P. Monceau for many useful discussions. The helpof Ž. Bihar with the plot in Fig. 1 is also acknowledged.References1) M. Ziman, Electrons and Phonons, Oxford (1963);2) H. B. G. Casimir, Physica 5 (1938) 495;3) P. Debye, Vorträge über die kinetische Theorie der Materie und der Elektrizität, Berlin(1914).4) R. Peierls, Ann. Physik 3 (1929) 1055;5) A. Matthiessen, Rep. Brit. Ass. 32 (1862) 144;6) C. Herring, Phys. Rev. 95 (1954) 954;7) P. G. Klemens, Proc. Roy. Soc. A 208 (1951).8) G. Leibfried and E. Schlömann, Nachr. Gött. Akad. 2a (1954) 71;9) J. C. Callaway, Phys. Rev. 113 (1959) 1046;10) A. A. Maradudin, J. Amer. Chem. Soc. 86 (1964) 3405;11) P. C. Kwok, Solid State Phys. 20 (1967) 213.FIZIKA A 1 (1992) 2, 175–180 179apostol: on the lattice thermoconductivity . . .O TOPLINSKOJ VODLJIVOSTI IDEALNOG KRISTALAMARIAN APOSTOLCentre de Recherches sur les Très Basses Températures, CNRS, BP 166 X.,38042 Grenoble-Cédex 9, FranceReceived 14 September 1992UDC 538.931Originalni znanstveni radOpisujući nosioce topline (Debyeve fonone) analogno ponašanju fotona dobivena jetemperaturna ovisnost toplinske vodljivosti rešetke u idealnom kristalu.180 FIZIKA A 1 (1992) 2, 175–180

O toplinskoj vodljivosti idealnog kristala

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