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The effect of uniaxial stress along the [001] direction on the scattering of phonons by bound acceptor holes in doped semiconductors has been investigated for the situations ω ωr where ωr is the resonance frequency. The relaxation rates for the off-resonance frequency regions are used in explaining anomalous resonance dips of the thermal conductivity curves of p-GaSb. Two conductivity integrals, K0 ωr are used. The present approach is quite successful in explaining the experimental results of thermal conductivity of p-type GaSb except at low temperatures. Es wird der Einflus einer uniaxialen Spannung in [001]-Richtung auf die Streuung von Phononen durch an Akzeptoren gebundene Locher in dotierten Halbleitern fur die Falle ω ωr, wobei ωr die Resonanzfrequenz ist, untersucht. Die Relaxationsraten fur die Bereiche auserhalb der Resonanzfrequenz werden benutzt, um die anomalen Resonanzminima der Kurve der thermischen Leitfahigkeit von p-GaSb zu erklaren. Zwei Leitfahigkeitsintegrale, K0 ωr benutzt werden. Das dargestellte Verfahren liefert auser bei niedrigen Temperaturen eine erfolgreiche Erklarung der experimentellen Ergebnisse der thermischen Leitfahigkeit von p-leitendem GaSb.

The resonant scattering of phonons in Sb-doped Ge is quantitatively explained on the basis of Kwok's theory as extended by Suzuki and Mikoshiba. These expressions are modified for inhomogeneously strained n-Ge and are used to explain the anomalous results of the strained sample for which 4Δ varies from (4Δ)min to (4Δ)max. The experimental results of the sample Sb-306, which is believed to be inhomogeneously strained, are explained with 4Δ1 = 0.1meV, 4Δ2 = 0.32 meV, and 4Δ3 = 0.42 meV with same sharing of the electron-phonon scattering rate such that the weighting factors are given by p1 = p2 = p3 = 1/3, where p1 + p2 + p3 = 1. Die Resonanzstreuung von Phononen in Sb-dotiertem Ge wird auf der Grundlage der von Suzuki und Mikoshiba erweiterten Kwokschen Theorie quantitative erklart. Diese Ausdrucke werden fur inhomogen verspanntes n-Ge modifiziert und zur Erklarung der anomalen Ergebnisse der verspannten Probe benutzt, fur die 4Δ sich von (4Δ)min auf (4Δ)max andert. Die experimentellen Ergebnisse der Probe Sb-306, von der angenommen wird, das sie inhomogen verspannt ist, werden mit 4Δ1 = 0.1meV, 4Δ2 = 0.32 meV, and 4Δ3 = 0.42 meV erklart mit der gleichen Verteilung der Elektron-Phonon-Streurate, die so ist, das die Gewichtsfaktoren durch p1 = p2 = p3 = 1/3, gegeben sind, wobei p1 = p2 = p3 = 1/3, ist.

It has been shown in the present work that the analytical expression of phonon-phonon scattering strength as obtained by Roufosse and Klemens together with an exponential factor can account for most of the observed phonon resistivity of Ge in both the high- and low-temperature regions. For the low-temperature region we have taken the results of Geballe and Hull for both the natural and isotopically enriched samples of Ge. For the high-temperature region we have considered the experimental results of Glassbrenner and Slack. As an additional example of the application of Klemens's expression, we have considered the phonon conductivity results for KCl in both the low- and high-temperature regions. The low-temperature results are taken from Seward's paper and the high-temperature data for KCl are from the measurements made by Devyatkova and Smirnov. For KCl, Klemens's expression, however, overestimates the three-phonon scattering strength by a factor which lies between 10 and 20 in both the low- and high-temperature regions. The form chosen for the phonon-phonon relaxation rate in the present paper has the merit of going smoothly from a correct high-temperature form to a form which approaches (but does not quite equal) the correct low-temperature form.

The resonance scattering of phonons in Co++ doped CdS is investigated both in the presence and absence of magnetic fields on the basis of modified expressions of Kozakov and Nagaev, which include weak resonance scattering of phonons for the off-resonance situations T ωr lead to plausible explanations for the unexplained depressions in the phonon conductivity of CaF2 in the temperature range 2 to 4 °K. Es wird die Resonanzstreuung von Phononen in mit Co++-dotiertem CdS sowohl bei Anwesenheit als auch bei Abwesenheit von magnetischen Feldern untersucht, und zwar auf der Grundlage von modifizierten Ausdrucken von Kozakov und Nagaev, die die schwache Resonanzstreuung von Phononen fur den Nichtresonanzfall T ωr fuhren zu plausiblen Erklarungen fur die unerklarten Ausdrucke in der Phononen-Leitfahigkeit von CaF2 im Temperaturbereich von 2 bis 4 °K.

The present work shows that the effect of the impurity-ion field on the electron-phonon interaction essentially needs consideration since it also allows the scattering of the phonons having wave vector $qg2{K}_{F}$, ${K}_{F}$ being the Fermi wave vector. Using Ziman's relaxation rate for the phonons $ql2{K}_{F}$, and Kosarev's relaxation rate for the phonons $qg2{K}_{F}$, we investigated whether or not the low-temperature lattice thermal-conductivity results in $n\ensuremath{-}\mathrm{G}\mathrm{e}$ may be explained with the same value of electron effective mass in the temperature range (1.5-6) \ifmmode^\circ\else\textdegree\fi{}K. However, one has to increase the product of the deformation potential and the density-of-states effective mass with the impurity concentration, which is well expected from the theory.

It is established in the present work that the decrease in low-temperature phonon conductivity of III-V semiconductors, caused by an introduction of impurities with concentration >or approximately=1017cm-3, may be explained by Kosarev's mechanism for the scattering of phonons by charge carriers in the effective electric field of the impurity centres for q>2kF, and a more consistent consideration of the screening of the phonon-electron interaction in the region of the long-wavelength phonons for Ziman scattering for q>2kF, as suggested by Crosby and Grenier (1971).

Expressions for resonance scattering of phonons by the bound donor electrons have been obtained for the first time for $n$-Ge in the presence of strains for the most general situation when, besides the usual valley-orbit splitting of the four-fold degenerate ground state into a singlet state and a triplet state, the triplet state splits into a doublet state and a singlet state due to internal static and dynamic strains present in the crystal. Similarly for $n$-Si, the most general situation is considered when the degeneracy of the donor-electron ground state is completely removed due to internal strains besides the usual valley-orbit splitting of the six-fold ground state into a singlet state and five-gold degenerate state. The expressions obtained for the resonance scattering relaxation rates of phonons by bound electrons for strained $n$-Ge are used to explain the phonon conductivity results of Sb-, P-, and As-doped Ge.

Low-temperature phonon conductivity results of compensated semiconductors such as Ge(P,Ga), Ge(As,Ga), and Ge(Sb,In) have been for the first time interpreted quantitatively on the basis of the theory proposed by Griffin and Carruthers and extended by Kwok. The compensated Ge samples are designated as Ge(donor impurity, compensating acceptor impurity). The basis of the present interpretations are the complete expressions obtained by Suzuki and Mikoshiba in the light of Kwok's theory for the resonance-scattering relaxation rates of phonons by bound donor electrons in Ge. The elastic and inelastic scatterings, both from the donor-electron singlet ground state and the next higher-energy triplet state, separated by $4\ensuremath{\Delta}$, as well as phonon-assisted absorption processes are considered in the present calculations in the temperature range 1-4\ifmmode^\circ\else\textdegree\fi{}K. The effect of compensation by acceptor impurities such as Ga in the case of P- and As-doped ($n$-type) Ge and by indium in the case of Sb-doped Ge is reflected in the variations of $4\ensuremath{\Delta}$ and the donor-electron radius ${a}_{0}$. Compensation in the case of P- and As-doped Ge by gallium reduces the values of $4\ensuremath{\Delta}$ but it almost remains unaffected in the case of Sb-doped Ge when it is compensated by indium. However, the donor-electron radius is reduced in all the cases on compensation.

The problem of three-phonon processes in solids is investigated on the assumption that all the phonons contribute equally to the conduction of lattice thermal waves in the sense that the deviations of their actual distributions from the equilibrium distributions are the same, i.e., ${\ensuremath{\psi}}^{\ensuremath{'}\ensuremath{'}}={\ensuremath{\psi}}^{\ensuremath{'}}=\ensuremath{\psi}$. The effects of normal and umklapp processes are separated by associating the heat flow with a drift distribution as well as a diffusive distribution. Choosing a form of the diffusive distribution, expressions for the normal and umklapp processes are derived. As a check of its validity the theory is then applied in the calculations of thermal conductivities of Ge, Si, KCl, and NaCl at high temperatures and the values so obtained are also compared with those obtained under the single-mode relaxation-time approximation. It is found that the present approximation ${\ensuremath{\psi}}^{\ensuremath{'}\ensuremath{'}}={\ensuremath{\psi}}^{\ensuremath{'}}=\ensuremath{\psi}$ gives a better estimate of the contribution of umklapp processes than the usual single-mode relaxation-time approximation ${\ensuremath{\psi}}^{\ensuremath{'}\ensuremath{'}}={\ensuremath{\psi}}^{\ensuremath{'}}=0$.

A simple method for the variational calculation of the thermal conductivity at very low temperatures is proposed in this paper. Isotropy and the absence of dispersion in the crystal vibrational spectrum are assumed. No distinction is made between transverse and longitudinal phonons. The assumed scattering mechanisms are (i) boundary scattering and (ii) mass-defect scattering. The trial function is chosen to be a power-series expansion in frequency. The thermal conductivity of germanium is calculated by applying the present method for the temperature range 2-20 \ifmmode^\circ\else\textdegree\fi{}K. At sufficiently low temperatures (2-10 \ifmmode^\circ\else\textdegree\fi{}K), the calculated results are in reasonably good agreement with the experimental results without using any adjustable parameter. For temperatures greater than 10 \ifmmode^\circ\else\textdegree\fi{}K, the calculated results show the presence of other scattering processes.

Temperature dependence of the first two bounds χ and χ on the thermal transport coefficient χ due to three-phonon umklapp processes is obtained. The results, both in the high-temperature and low-temperature approximations, are in agreement with Benin's calculations for solid argon. Es wird die Temperaturabhangigkeit der ersten beiden Grenzen χ und χ vom thermischen Transportkoeffizienten χ durch drei-Phononen Umklappprozesse erhalten. Die Ergebnisse befinden sich sowohl in der Niedertemperatur- als auch in der Hochtemperatur-naherung mit den Berechnungen von Benin fur festes Argon in Ubereinstimmung.

The Boltzmann transport equation is solved for phonons in an impure insulator slab to obtain the simplest possible and physically plausible solution. The equation is not treated as a boundary value problem, but it is assumed that the occupation number for phonon states corresponding to a given frequency and polarization is a smooth function of the direction of phonon wave vector. Heat flux and energy density in the crystal are studied when such a solution is assumed to exist. Die Boltzmannsche Transportgleichung wird fur Phononen in einem verunreinigten Isolatorstab berechnet, um die einfachst mogliche und physikalisch plausible Losung zu bekommen. Die Gleichung wird nicht als ein Grenzwertproblem behandelt, sondern es wird angenommen, das die Besetzungszahlen fur Phononenzustande, die mit einer gegebenen Frequenz und Polarisation ubereinstimmen, eine glatte Funktion der Richtung des Wellenvektors der Phononen ist. Warmeflus und Energiedichte im Kristall werden unter der Annahme, das eine solche Losung existiert, untersucht.

Study has been made of the effect of acoustic power applied at twice the Larmor frequency on the individual nuclear magnetic resonance lines in the presence of radio-frequency power. When\(W''_{u2} (W_{u\tfrac{1}{2} \to - \tfrac{3}{2}} )\) is finite and other ultrasonic transition probabilities are zero, one obtains the same acoustic saturation behaviour for the central resonance line for the quadrupolar and the magnetic relaxations. For an arbitrary spinI, this expression has been generalized. However, the central resonance line remains unaffected when ultrasonic transition probabilities other than\(W''_{u2} \) are finite separately. The effect of the acoustic power on the satellites for different cases is studied in great detail. Some of the satellites are affected and a few are unaffected. Some of the affected satellites show interesting behaviour in the sense that, instead of the regular saturation behaviour, enhancement effect and negative ratios of the population difference to its equilibrium value are obtained for higher values ofU which is a measure of acoustic power.

A variational method is used to calculate the three-phonon umklapp resistivity in Ge, Si, NaCl, and KCl where the expressions are used for the three-phonon scattering strengths, derived previously for a simple cubic crystal. The temperature ranges, considered for Ge, Si, NaCl, and KCl are 400 to 1000 °K, 400 to 1600 °K, 30 to 80 °K, and 40 to 100 °K, respectively. The present calculations show that for the semiconductors Si and Ge the calculated umklapp resistivities are half as great as the experimentally observed values. This suggests that the three-phonon normal processes are quite important for Si and Ge even for temperatures greater than 400 °K. For alkali halides the agreement between theory and experiment is reasonably good. It is also found that the contribution due to T + T ⇌ L processes towards the umklapp resistivity is greater than that of the T + L ⇌ L processes for all the materials at all temperatures. Es wird eine Variationsmethode zur Berechnung des Drei-Phonon-Umklapp-Widerstan-des in Ge, Si, NaCl und KCl benutzt. Bei diesen Berechnungen wurden die Ausdrucke fur die Drei-Phononen-Streustarken, die bereits frEaher abgeleitet wurden, far einen einfach kubischen Kristall benutzt. Die beracksichtigten Temperaturbereiche fur Ge, Si, NaCl und KCl waren 400 bis 1000 °K, 400 bis 1600 °K, 30 bis 80 °K bzw. 40 bis 100 °K. Die vorgelegten Berechnungen zeigen, das fur die Halbleiter Si und Ge die berechneten Umklapp-Widerstande halb so gros sind wie die experimentell beobachteten Werte. Dies weist darauf hin, das die normalen Drei-Phononprozesse fur Si und Ge sogar fur Temperaturen uber 400 °K ganz wesentlich sind. Fur die Alkalihalogenide ist die Ubereinstimmung zwischen Theorie und Experiment ziemlich gut. Es wird auch gefunden, das der durch T + T ⇌ L-Prozesse hervorgerufene Beitrag zum Umklapp-Widerstand groser ist als der der T + L ⇌ L-Prozesse fur alle Materialien bei allen Temperaturen.

Huebener’s analysis of the influence of point defects on the phonon-drag thermoelectric power is extended to include the effects of electron-phonon normal scattering processes and the results are applied to thermoelectric power of quenched-in lattice vacancies of gold. The present analysis reveals that the frequency-independent scattering parameter for the Rayleigh-type scattering of phonons due to quenchedin lattice vacancies is temperature-dependent.

It is shown that one can explain the temperature dependence of the phonon conductivity of Ge in the entire range from 2 to 1000 \ifmmode^\circ\else\textdegree\fi{}K if the three-phonon relaxation rate is given by $\ensuremath{\tau}_{3\mathrm{ph}}^{\ensuremath{-}1}\ensuremath{\propto}g(\ensuremath{\omega}){T}^{m(T)}{e}^{\ensuremath{-}\frac{\ensuremath{\Theta}}{\ensuremath{\alpha}T}}$. Three-phonon scattering processes are classified, after Guthrie, into two groups: class I, which involves the annihilation of carrier phonons by combination, and class II, which involves splitting of carrier phonons. At all temperatures, the values of $m(T)$ for both classes of processes lie either definitely below or close to the upper limit of $m(T)$ as obtained by Guthrie.

The recent modification of Holland's model of two-mode conduction, as proposed by us, has been applied to explain the phonon conductivity of InSb and GaAs. This model, Known as the Sharma-Dubey-Verma model, makes use of Guthrie's classification of three-phonon scattering events. In this model, the exponent $m$ of the temperature, i.e., ${T}^{m(T)}$ is a continuous function of temperature and approaches unity in the high-temperature region for both the longitudinal phonons as well as transverse phonons. The dispersion of acoustic branches is taken into account in replacing $\frac{{v}_{g}}{v_{p}^{2}}$ in the conductivity integrals and this forms the basis of the division of the conductivity integrals for the different polarization branches. The present model gives excellent agreement between the theoretical and experimental values of phonon conductivity except near the maximum where the scattering of phonons by point defects dominates over phonon-phonon scattering as well as boundary scattering of phonons.

Various theories connected with ultrasonic investigations of nuclear spin-lattice relaxation are presented. The effects of covalency on electrostatic interaction energy of a nucleus with the charges constituting its environment are proposed in theoretical calculations. A mathematical treatment is contained on the probability per unit time that sound waves will produce a transition of a nuclear spin between its Zeeman levels. Expressions are also presented on equilibrium nuclear magnetization in the presence of ultrasonic and thermal lattice vibrations. Experimental ultrasonic techniques based on the pulsed nuclear induction method of measuring nuclear magnetization are described.

The electronic polar and dipolar thermal conductivity of both $n$- and $p$-type Si-Ge alloys with different carrier concentrations are calculated in the temperature range 300 to 1100\ifmmode^\circ\else\textdegree\fi{}K. With carrier concentration remaining constant, the nature of the scattering mechanism is determined from the temperature dependence of the conductivity mobility. The scattering parameter being known, the temperature dependences of the reduced Fermi level and the density-of-states effective mass are determined from thermoelectric measurements on these samples by Dismukes et al. These temperature variations are taken into account in the calculation of the contribution of the electron-phonon interaction to the thermal resistance of the doped Si-Ge alloys. The thermal conductivities due to longitudinal phonons and transverse phonons are calculated separately, and there is good agreement between the calculated temperature dependences of the total phonon conductivity and the experimentally obtained values of thermal conductivity minus the electronic thermal conductivity. The dilatational deformation potential is found to increase with the increase in the carrier concentration and the reduced Fermi potential. The dipolar contribution is found to be significant only at temperatures above 700\ifmmode^\circ\else\textdegree\fi{}K and for $n$-type alloys, for which the doping is comparatively low.

Introducing a parameter a which determines the relative strength of normal and umklapp phonon processes, the correction to phonon conductivity of undoped and doped Ge due to normal processes has been estimated in the framework of Callaway's model in the temperature range from 2 to 80 K for various values of a ranging from 0.1 to 10. There is an overall increase in the correction term with the increase in temperature, the maximum being below 10%.

The theoretical models of Callaway and Klemens are compared to experimental results in (a) silicon and diamond, (b) solidified argon and neon, and (c) KCl and NaF, which are taken from the literature. Three sources of scattering are postulated: phonon-phonon processes, isotopic point defect scattering, and boundary scattering, except in NaF where isotopic scattering is absent. The Callaway model, in all cases, gives a good fit and excellent agreement is obtained for diamond and solidified argon and neon. In the vicinity of the conductivity maxima the Callaway model gives a better fit with the experimental curve. The disagreement between the results of the two models is most pronounced in solidified argon and neon where the Debye temperature is low and Umklapp processes are predominant at the conductivity maximum. In diamond, where the Debye temperature is quite high and Umklapp processes are unimportant at the temperatures of interest the results of both models are more or less similar. The Casimir model for finding the characteristic length for the boundary scattering fails to explain the observed results in solidified argon and neon. This indicates the existence of microscale fluctuations in the composition of the solid. In NaF the characteristic length has to be decreased by a factor of about 5. For diamond the value of the parameter $A$, which gives the best fit with experiment, is a hundred times the value calculated from Klemens' expression for $A$. This supports the earlier suggestion that there occur clusters of point defects in diamond. Good agreement with the experimental results beyond the conductivity maximum is obtained by taking (${B}_{1}+{B}_{2}$) to be temperature independent in all the cases.

Acoustic absorption measurements by the pulse technique have been made in methyl-alcohol-cyclohexane critical mixtures from 08-25 MHz and from 37 to 55°C (critical temperature 491 °C). The dependence of absorption on frequency and temperature is found to be in only qualitative agreement with the predictions of Fixman's theory.

Temperature dependence of the density-of-states effective mass is obtained in the case of the heavily doped semiconductors for which the impurity levels merge with the conduction band and Ziman scattering of phonons by conduction electrons is the relevant scattering. Phonon conductivity results of Sb- and As-doped Ge are used to obtain the values of m∗ in the range 1 to 20 K.

The simplified variational method has been extended to Ziman's electron-phonon scattering and the results are applied to explain the phonon conductivity results of heavily doped n-Ge, for which donor-electron states merge with conduction states and phonons are scattered by donor electrons in the conduction state. The expressions for the scattering matrix elements for the electron-phonon scattering are obtained and its contribution to the thermal resistivity is considered in the temperature range 2 to 10 °K in the presence of boundary scattering and isotopic scattering of phonons. The analysis of the thermal resistivity data reveals that the effective mass of the donor electron in the conduction state varies from 0.25 to 0.45 in the temperature range 2 to 10 °K. Es wurde die vereinfachte Variationsmethode auf die Zimansche Elektron-Phonon-Streuung ausgedehnt und die Ergebnisse zur Erklarung der Phononenleitfahigkeit von stark dotiertem n-Ge verwendet, fur das die Donator-Elektronenzustande mit den Leitungsbandzustanden uberlappen und die Phononen durch Donatorelektronen in Leitungszustanden gestreut werden. Die Ausdrucke fur die Streumatrixelemente fur die Elektron-Phonon-Streuung werden erhalten und deren Beitrag zum thermischen Widerstand im Temperaturbereich 2 bis 10 °K bei Anwesenheit von Grenzflachenstreuung und Isotopenstreuung von Phononen berucksichtigt. Die Analyse der thermischen Widerstandswerte zeigt, das die effektive Masse des Donatorelektrons im Leitungszustand sich von 0,25 auf 0,45 im Temperaturbereich von 2 bis 10 °K andert.

Anderson's theory has been used to explain the temperature dependence of the adiabatic bulk modulus of KCl, NaCl, LiF and TlBr in the temperature range up to the melting point of the material. In KCl and LiF, the value of δ which gives good agreement between theory and experiment is that which is obtained from the pressure dependence of B s . In NaCl the value of δ for the best fit is the one obtained from the temperature coefficient of ln B s . In TlBr Chang's relation δ = 2 γ , however, gives a better agreement between the observed temperature dependence of B s and the theoretical values.

Expressions have been derived for magnon-phonon scattering relaxation times for different cases as a function of temperature and wave vector using the theory of Sinha and Upadhyaya. The appropriate relaxation time is used in the determination of the effective relaxation time for magnons, and the thermal conductivity of yttrium iron garnet has been calculated in the temperature range 0.5 to 20\ifmmode^\circ\else\textdegree\fi{}K using a quadratic dispersion law for magnons and a Debye spectrum for phonons.

The reduction in the thermal conductivity of Te-doped GaSb samples at temperatures below the maximum conductivity temperature is explained in detail by considering the scattering of phonons by conduction electrons. The Ziman model has been used to fit the phonon-conductivity results on Te-doped GaSb samples with excess donor-electron concentrations in the range 1.6-1.8 \ifmmode\times\else\texttimes\fi{} ${10}^{17}$ ${\mathrm{cm}}^{\ensuremath{-}3}$, where the impurity levels merge with the conduction band. At low concentrations of the order 1.6 \ifmmode\times\else\texttimes\fi{} ${10}^{17}$ ${\mathrm{cm}}^{\ensuremath{-}3}$ the phonons are scattered by electrons in the 000g minima band and at concentrations exceeding 1 \ifmmode\times\else\texttimes\fi{} ${10}^{18}$ ${\mathrm{cm}}^{\ensuremath{-}3}$ the phonons are scattered by electrons in the 111g minima band. At intermediate concentrations phonons are scattered by electrons in both the bands.

The theoretical model of Callaway for lattice thermal conductivity at low temperatures is compared with the experimental results in ZnF 2 , CaF 2 , PbS and InSb. Good agreement between the theory and experiment is obtained in all the cases. The present analysis of the results in ZnF 2 and PbS indicate the presence of microscale fluctuations in the composition of the solid. In CaF 2 and PbS the impurity scattering is predominant. In InSb boundary scattering is negligible in the temperature range in which the data are available.

A phenomenological model is proposed within the framework of BCS theory and the Boltzmann equation approach for the electronic thermal conductivity of superconductors limited by phonon scattering. Values of the parameters of the model are determined from comparison with the experimental data. In the case of weak coupling of electrons and lattice vibrations, it is found that ${\ensuremath{\tau}}_{s}$, the time a quasiparticle takes to relax to the equilibrium flow of supercurrent, is comparable with ${\ensuremath{\tau}}_{n}$ in the normal state. In the case of strong electron-phonon interaction, it is found that ${\ensuremath{\tau}}_{s}$ is ${10}^{14}$ times ${\ensuremath{\tau}}_{n}$ at the reduced temperature $t=0.80$.

Reformulation of the lattice thermal conductivity of solids has been given, using the fact that the three-phonon umklapp processes can be divided into different classes depending on the reciprocal-lattice vectors of the crystal concerned, and that they tend towards displaced distribution functions characteristic of their own. The expression obtained differs from that of Callaway by terms which are expected to be of minor importance in real systems.

A phenomenological form for $\ensuremath{\omega}$ dependence of ${\ensuremath{\tau}}_{r}^{\ensuremath{-}1}$ is proposed which takes into account a term known as the population-difference factor. The thermal-conductivity-vs-temperature curves for CdTe: ${\mathrm{Fe}}^{+2}$ and MgO: ${\mathrm{Fe}}^{+2}$ are explained satisfactorily except that there are slight deviations at the low-temperature as well as at the high-temperature side.

Keating's theory is applied to explain the recent experimental data of the third‐order elastic moduli for Ge at room and liquid‐nitrogen temperatures. The agreement between theory and experiment obtained for the new data at 293°K is better than that obtained by Keating. Also, the theory explains the experimental data at 77°K in an excellent way. It is found that unlike the other anharmonic interactions, the nearest‐neighbor central one is further enhanced with the lowering of the temperature. The important parameter in Akhiezer's theory for microwave phonon attenuation in solids, called the nonlinearity constant D, has also been calculated for the said temperatures for 〈100〉 and 〈111〉 directions of ultrasonic longitudinal wave propagation, and in both the cases, it is found to be temperature dependent. The present work represents the first experimental support for our previous prediction, derived on the phonon conductivity data for Si, that D is temperature dependent and decreases at low temperatures.

It is shown that the high-temperature behavior of the lattice thermal conductivity of silicon, which has been carefully remeasured recently by Fulkerson et al., can be explained by the four-phonon processes. It has been also found that the major contribution to the phonon thermal conductivity at high temperatures comes from the high-frequency transverse phonons.