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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 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.

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.

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.

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.

A pulse comparison method has been used to measure ultrasonic absorption in mixtures of glycerol and water in the temperature range 25 to 60 °C at 0.8 and 5 MHz. The behaviour at 5 MHz is similar to what has been observed by Willis. At 0.8 MHz α/f 2 increases with the increase in temperature for concentrations up to 08% glycerol in water and shows a maximum at about 20% of water in glycerol at and above 50 °C. The height of the maximum increases with the increase in temperature. A similar maximum is observed in ultrasonic velocity. The present results are interpreted on the basis of an additional slow relaxational absorption mechanism such thatωτ ≫ 1 at 0.8 MHz. The maximum in the absorption and velocity corresponds to the formation of a complex of the type (CH2OH-CHOH-OH2OH)4 H2O, the formation of which is favoured at high temperatures. The dissolution and formation of this complex provides the additional relaxational mechanism.

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.

Measurements of ultra-sonic absorption at different temperatures in the mixtures of glycerol and dioxane below 1 MHz using the pulse-comparison method reveal the presence of hydrogen-bonded complexes. The observed excess absorption goes through peaks at 0.21 and 0.81 mole fractions of glycerol thereby indicating the compositions of the complexes to be AB4 and A4B, where A and B refer to glycerol and dioxane molecules respectively. The present results have been explained in terms of the formation and dissolution of the above complexes leading to relaxation mechanisms such thatωτ ≪ 1. The formation of hydrogen bonded complexes is further supported by the calculation of the free-energy difference ΔF between the two states. Measurements of ultra-sonic velocity in the mixtures using the total-reflection technique also show a slight hump near the concentration at which the first absorption peak is formed. The formation of bonds in the complexes is also discussed.

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 measurements of the ultrasonic absorption and velocity in the mixtures of glycerol and water at 0.8 and 0.5 MHz reveal for the first time the formation of the complexes between the molecules of glycerol and water of the type AB and AB6, where A and B refer to glycerol and water molecules. The formation of the complexes is further supported by the calculation of the free-energy difference ΔF between the two states which correspond to absence and presence of the complexes. The relaxation frequency of the reaction mechanism AA+BB⇆2 AB varies from 0.815 kHz at 40°C to 2.707 kHz at 60°C. The relaxation frequency of the process A+B6⇆AB6 varies from 0.6 kHz at 40°C to 1.407 kHz at 60°C. The relaxation frequencies of the latter process are in general lower than those of the former.

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 $t=0.80$.

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.

The contribution of magnons to thermoelectric power may be appreciable in ferromagnets at sufficiently low temperatures. In this paper an analysis of the magnon-drag thermoelectric power has been given and a qualitative estimate of magnon contribution to the thermoelectric power and the effect of magnetic field has been obtained.

The drastic reduction in the phonon conductivity of P-doped Ge samples, as observed by Mathur and Pearlman in the temperature range (1.4-5) \ifmmode^\circ\else\textdegree\fi{}K, can be explained quantitatively in terms of the scattering of phonons by electrons in bound states as well as in conduction states, depending upon the donor-electron concentration of the sample. For electron concentrations ${N}_{\mathrm{ex}}l{10}^{17}$ ${\mathrm{cm}}^{\ensuremath{-}3}$, phonons are scattered by bound donor electrons. The modified expressions of Kumar, Srivastava, and Verma for the electron-phonon---scattering relaxation rate, based on Kwok's theory, explain the phonon conductivity results very well. The adjusted values of the shear deformation potential for the best fit between theory and experiment for different P-doped Ge samples are of the order of 13.32-14.18 eV, which agrees very well with the usual experimental values of 15-19 eV. For ${N}_{\mathrm{ex}}g{10}^{17}$ ${\mathrm{cm}}^{\ensuremath{-}3}$, impurity levels merge into the conduction band and Ziman's theory of phonon scattering by conduction electrons is the relevant phonon scattering mechanism. The values of the longitudinal deformation potential for different P-doped Ge samples lie in the range - 1.5 to - 5.07 eV. Values of the density-of-states effective mass ${m}^{*}$ have been determined for different samples in the temperature range (1-5) \ifmmode^\circ\else\textdegree\fi{}K. The term $\frac{{m}^{*}}{m}$ increases with temperature and its range of variation is from 0.22 to 0.60.

Optimized values of the valley-orbit splitting $4\ensuremath{\Delta}$ and the donor-electron radius ${a}_{0}$ for the different Sb-, As-, and P-doped Ge samples are obtained from the analysis of the resonance scattering of phonons which play a dominant role in the phonon resistivity of these materials in the liquid-helium range. Both elastic and inelastic scatterings from the singlet and triplet states as well as phonon-assisted absorption processes are considered in the calculations for the entire resonance region. Excellent agreement between theory and experiment is obtained for the optimized values of ${a}_{0}$ and $4\ensuremath{\Delta}$.

In the present work it has been shown that the discrepancy between Kwok's theory and the observed temperature dependence of acoustic-phonon attenuation in P-doped Ge beyond its maximum can be partially explained if one modifies Kwok's theory to include the transitions to singlet and triplet states from higher-energy states which may be relevant at higher temperatures. This work confirms that Kwok's theory overestimates phonon attenuation in the higher-temperature region.

An explanation is provided for the thermal-conductivity results of antiferromagnetic Fe${\mathrm{Cl}}_{2}$ (${T}_{N}=23.5\ifmmode^\circ\else\textdegree\fi{}$K) in the temperature range 3-80 \ifmmode^\circ\else\textdegree\fi{}K in terms of the phonon-induced electronic excitations among the trigonal crystal and exchange-field-split energy levels of the ${\mathrm{Fe}}^{2+}$ ions in the crystalline lattice of Fe${\mathrm{Cl}}_{2}$. The phonon scattering is described by a relaxation term which has a resonant character. The boundary, point-defect, and the umklapp scattering processes are basic to both Fe${\mathrm{Cl}}_{2}$ and Cd${\mathrm{Cl}}_{2}$. In nonmagnetic Cd${\mathrm{Cl}}_{2}$, they are, moreover, found sufficient to account for the experimental results.

The present work shows that one can explain the phonon-conductivity results of $n\ensuremath{-}\mathrm{Ge}$ in the temperature range 0.3-4.2\ifmmode^\circ\else\textdegree\fi{}K on the basis of more exact theory of the scattering of phonons by bound electrons as given by Kwok and extended by Kumar, Srivastava, and Verma, which includes inelastic scattering of phonons as well as the scattering off the triplet state.

In this paper we have discussed the variational treatment of the boundary scattering of phonons. The procedure of Hamilton and Parrott for boundary scattering of phonons is modified; but still the method is shown to give Casimir's result at low temperatures.

Klein's theory of the resonant phonon scattering from a two-level tunneling system has been applied to the KCl: ${\mathrm{CN}}^{\ensuremath{-}}$ system. Unlike the KCl: ${\mathrm{OH}}^{\ensuremath{-}}$ and NaCl: ${\mathrm{OH}}^{\ensuremath{-}}$ systems, agreement between theory and experiment is achieved with $n=1$. It is concluded that the scattering due to the mass defect ($n=2$) and the scattering due to force-constant changes ($n=4$) do not play a significant role in causing the resonance dips in the $K$-vs-$T$ curves in KCl: ${\mathrm{CN}}^{\ensuremath{-}}$ systems.

A variational treatment is used to calculate the contributions of longitudinal and transverse phonons towards the phonon conductivity of Ge in the Ziman limit. The matrix elements occurring in the formula for thermal conductivity are evaluated using the corrected expressions of three-phonon scattering strengths.