The behaviour of the impurity modes due to a pair of substitutional impurities characterized by both mass as well as force-constant changes and occupying nearest neighbour positions in a diatomic linear chain, is studied. The results are compared with those for the case of impurity pairs occupying next nearest neighbour sites discussed earlier as well as the existing three dimensional calculations of Elliott and Pfeuty. The nearest neighbour impurity pair gap and local modes can be interpreted in terms of two single impurities substituted in the two different sublattices unlike the next nearest neighbour pair modes. The inband resonant modes are totally new features characteristic of the pair. Finally, the predictions of the theory are compared with the available experimental data for Si-impurity-pair-complexes and qualitative agreement is shown.

Lucovsky, Brodsky, Burstein (LBB) have studied the behaviour of mixed crystals by setting up a criterion for the existence of local mode frequencies in real crystals starting from a diatomic linear chain model. This, while successfully predicting the one and two mode behaviour for some systems fails to predict the mixed mode behaviour. We propose a similar criterion for the existence of gap modes, by demanding that the gap mode predicted by the diatomic linear chain model should lie within the gap of the real three dimensional solid for its existence. It is shown that the gap modes for various systems calculated using this criterion are in reasonable agreement with the experimental values. The infrared behaviour of mixed crystals has to be determined by examining the existence of local as well as gap modes for the two end members of the system. This generalized new criterion successfully predicts the mixed mode behaviour of III–V mixed crystals besides predicting the one and two mode behaviour, observed in infrared absorption of mixed alkali halioes and III-V compounds.

The modes of vibration of a three impurity cluster in a diatomic linear chain is studied. The configuration of the three impurity atoms is such that two of these go into next nearest neighbour sites in one of the sub-lattices whereas the third occupies the central in between site in the other sub-lattice. New local, gap and inband modes are seen to exist, besides the ones corresponding to the impurity pairs. A new feature which appears is that the frequencies of some of the pair modes remain totally unaffected in the presence of the third impurity atom.

A general solution is obtained for the lattice dynamics of a cluster ofn-impurity atoms using the double-time Green’s function formalism. The cluster is characterized byn-mass defect andm-force constant change parameters. It is shown that this general solution for the Green’s function for then-impurity cluster can also be expressed in terms of the Green’s function for the (n−1)-impurity cluster. As an application, the cluster impurity modes for a pair are calculated using the Debye model for the host lattice dynamics. The splitting of the high frequency local modes and nearly zero frequency resonant modes due to pairs show an oscillatory behaviour on varying the distance of separation between the two impurity atoms. These oscillations are most prominent for two similar impurities and get damped for two dissimilar impurities or if one of the impurities produces a force constant change. The predictions of the calculation provide qualitative explanation of the data obtained from the infrared measurements of the resonant modes in mixed crystal system of KBr_1−c Cl_c: Li⁺ and KBr_1−c I_c: Li⁺.

Exact solutions for the motion of a classical anharmonic oscillator in the potentialV(φ)=Bφ² − |A|φ⁴ +Cφ⁶ are obtained in (1 + 1) dimensions. Instanton-like solutions in (imaginary time) which takes the particle from one maximum of the potential to the other are obtained in addition to the usual oscillatory solutions. The energy dependence of the frequencies of oscillation is discussed in detail. This can be used as a model for the first order structural phase transition in the mean field approximation. The high and low temperature behaviour of the static susceptibility is obtained. Finally, a qualitative explanation is offered for the observed central peak in ferroelectrics like SrTiO₂.

The classical φ⁶-field theory in (1+1) dimensions, is considered as a model for the first order structural phase transitions. The equation of motion is solved exactly; and the presence of domain wall (kink) solutions at and below the transition point, in addition to the usual phonon-like oscillatory solutions, is demonstrated. The domain wall solutions are shown to be stable, and their mass and energies are calculated. Above the transition point there exists exotic unstable kink-like solutions which takes the particle from one hill top to the other of the potential. The partition function of the system is calculated exactly using the functional integral method together with the transfer matrix techniques which necessitates the determination of the eigenvalues of a Schrödinger-like equation. Thus the exact free energy is evaluated which in the low temperature limit has a phonon part and a contribution coming from the domain wall excitations. It was shown that this domain wall free energy differs from that calculated by the use of the domain wall phenomenology proposed by Krumhansl and Schrieffer. The exact solutions of the Schrödinger-like equation are also used to evaluate the displacement-displacement, intensity-intensity correlation functions and the probability distribution function. These results are compared with those obtained from the phenomenology as well as the φ⁴-field theory. A qualitative picture of the central peak observed in structural phase transitions is proposed.

The question of the occurrence of two-phonon bound states in imperfect crystals is investigated. It is shown that the anharmonicity mediated two-phonon bound state which is present in perfect crystals gets modified due to the presence of impurities. Moreover, the possibility of the occurrence of a purely impurity mediated two-phonon bound state is demonstrated. The bound state frequencies are calculated using the simple Einstein oscillator model for the host phonons. The two-phonon density of states for the imperfect crystal thus obtained has peaks at the combination and difference frequencies of two host phonons besides the peaks at the bound state frequencies. For a perfect crystal the theory predicts a single peak at the two-phonon bound state frequency in conformity with experimental observations and other theoretical calculations. Experimental data on the two-phonon infrared absorption and Raman scattering from mixed crystals of GA_1−c Al_c P and Ge_1−c Si_c are analysed to provide evidence in support of impurity-mediated two-phonon bound states. The relevance of the zero frequency (difference spectrum) peak to the central peak observed in structural phase transitions, is conjectured;

The existence of a domain wall-like contribution to the free energy above the first order phase transition point is demonstrated for a system described by the ϕ⁶-field theory in (1+1) dimensions.

The concentration dependence of the reststrahl absorption in various mixed crystals exhibiting one, two and mixed mode behaviour is investigated using the coherent potential approximation (cpa). The phonon Green’s function, the impurity mode frequencies and the strength of absorption are calculated in the Einstein model from the generalizedcpa proposed by Tripathi and Behera which takes into account both mass and force constant changes. The introduction of a phenomenological concentration dependence of the force constant change parameter is shown to provide a satisfactory explanation of the concentration dependence of the experimental data for the twenty mixed crystal systems analysed. It is conjectured that the nearest neighbour force constant of an impurity atom substituted at a host site is very much different from that of a perfect crystal consisting of these impurity atoms and that both these play an important role in determining the one, two and mixed mode behaviour of the mixed crystals.

A coherent potential approximation (cpa) for a mixed diatomic linear chain including both mass and force constant changes has been developed. In this case an impurity atom substituted at a particular site in one of the sublattices couples with two nearest neighbour atoms in the other sublattices. The diatomic linear chain is therefore considered as a tetratomic linear chain, the size of the unit cell being twice the original. Thecpa density of states and the dielectric susceptibility have been calculated. The numerical values of the later have been calculated in theata (averaget-matrix approximation) limit. Comparison of these results with the experimental and other computer calculations show a qualitative agreement.

The effect of small concentrations of non-magnetic substitutional impurities (characterized by changes in mass, nearest neighbour force constants and local electron-phonon interaction) on the observability of superconducing (sc) gap excitations in 2H-NbSe₂ by Raman scattering is considered within the theoretical framework of Balseiro and Falicov (bf). The phonon self-energy when evaluated forq=0 using the Einstein oscillator model for the host phonons is seen to have a pole corresponding to the impurity mode besides the gap excitation one around 2Δ previously obtained bybf. A splitting of the impurity mode into two in thesc state is predicted. An enhancement occurs in the strength of thesc gap excitation peak whenever it is close to the impurity mode. The data from the experiments of Sooryakumar, Klein and Frindt on impure samples of 2H-NbSe₂ are discussed in the light of the present calculation.

A theoretical model with electron-phonon and anharmonic interactions is proposed to explain the two-phonon mode observed in the Raman spectra of layered transition metal dichalcogenides, which exhibit charge density wave (cdw) phase transition. The phonon self-energy, which involves the electron response function and the two-phonon Green’s function, is calculated using the double-time Green’s function formalism. It is shown that in these low-dimensional systems there exists an anharmonicity-mediated two-phonon mode in the phonon spectral function both in the normal and in thecdw phases. In the normal phase since the phonon Raman scattering proceeds through a single optic phonon the calculations are carried out for zero wave vector and hence the contribution of the electron response function to the self-energy vanishes. On the other hand explicit evaluation of the two-phonon Green’s function shows that the frequency of the two-phonon mode is twice that of the Kohn anomaly phonon and decreases with decreasing temperature. The strength of two-phonon peak is found to be comparable to that of the original optic phonon. In thecdw phase the phonon which enters into the Raman scattering is taken to be the one with thecdw wave vectorQ, which when zone-folded becomes equivalent to zero wave vector. The evaluation of the electron response function in this phase generates a phonon corresponding to thecdw-amplitude mode. The two-phonon Green’s function is assumed to be of similar form as in the normal phase. The spectral function evaluated at zero temperature shows a weak two-phonon peak besides the prominentcdw-amplitude mode. Numerical results are presented for the system 2H-NbSe₂ and are found to be in qualitative agreement with the experimental data.

The electron-phonon interaction in the periodic Anderson model (PAM) is considered. The PAM incorporates the effect of onsite Coulomb interaction (U) between f-electrons. The influence of Coulomb correlation U on the phonon response of the system is studied by evaluating the phonon spectral function for various parameters of the model. The numerical evaluation of the spectral function is carried out in the long wavelength limit at finite temperatures keeping only linear terms in U. The observed behaviour is found to agree well with the general features obtained experimentally for some heavy fermion (HF) systems.