• K S Viswanathan

      Articles written in Proceedings – Section A

    • On the characteristic vibrations of linear lattices

      K S Viswanathan

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      The nature of wave propagation in a linear lattice consisting ofp particles in its unit cell has been studied and it is shown that the group velocity of the waves corresponding to each of the (2p—1) characteristic frequencies of the lattice is identically equal to zero. It has been further proved that the state of movements of the particles resulting from an initial disturbance localised in a finite region of the lattice tends asymptotically to a superposition of the (2p—1) characteristic vibrations of the lattice. In (p—1) modes of these vibrations, equivalent particles in successive cells vibrate with the same phases and amplitudes, while in the ramainingp normal modes, vibration occurs with equivalent particles moving alternately with opposite phases. These results from an one-dimensional analogue of the general theory of vibrations of crystal lattices formulated by Sir C. V. Raman and are generalisations of the results corresponding to the casesp=1 andp=2 which were first obtained respectively by Hamilton and by Nagerdra Nath and S. K. Roy. In view of the decay of the anplitudes of vibrations of the particles with time, these results hold good for a finite lattice also, provided its length is sufficiently large and the domain of initial disturbance is far from its ends.

    • The characteristic vibrations of a rectangular lattice

      K S Viswanathan

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      For a rectangular lattice with one particle in each unit cell, it is shown that the group velocity of the waves vanishes for the six characteristic frequencies and that the state of movements of the particles arising out of an initial disturbance tends to a superposition of these six characteristic vibrations of the lattice. These six frequencies would reduce to three for a square lattice on account of its symmetry; in all these two cases however, the amplitudes of vibrations of the particles vary inversely as the time elapsed from the instant of the initial disturbance. The physical interpretation of these results and their applicability to the case of a finite lattice are discussed.

    • The characteristic vibrations of crystal lattices—Part I

      K S Viswanathan

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    • The characteristic vibrations of crystal lattices—Part II

      K S Viswanathan

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      It is shown that the wave and group velocities of the elastic waves are identical. An initial disturbance consisting of a translation of a supercell excites asymptotically the (3p–3) characteristic vibrations only, the amplitudes of the remaining 21p modes being zero to a first order of approximation. A translation of a single unit cell results in exciting all the (24p–3) modes of vibrations of the crystal. The question of reduction of the asymptotic expression for the displacements of the atoms about their mean positions to its simplest form, when degeneracies are present in these vibrations, is also considered.

    • The theory of elasticity and of wave-propagation in crystals from the atomistic standpoint

      K S Viswanathan

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      The static method of obtaining the strain energy function of a crystal has been developed for the case where the potential energy of the entire lattice is a general quadratic in the nuclear displacements of the atoms of the crystal. It is shown that for heterogeneous strains, the deformation energy of the crystal is a quadratic in all the nine strain components. For small homogeneous deformations, the strain energy function reduces to the form of the corresponding function in the elasticity theory. The wave equations obtained from this energy function by the variational procedure are identical with the wave equations of the elasticity theory. By comparing either the two forms of the strain energy function or the two sets of waveequations, expressions for the elastic constants can be obtained in terms of the atomic force constants.

      Starting from the most general expression for the strain energy function which is a quadratic in all the nine strain components and by assuming that the strain components are all linearly independent functions of the position vector of any point of the solid, the wave equations of Begbie and Born have been derived by means of Hamilton’s variational principle. But these equations are not reducible to the symmetric form of the wave equations of the elasticity theory without further assumption of additional relations among the force constants. Since there is no justification for such new relations which restrict the generality of the force scheme used, the expressions of Begbie-Born and of Kun Huang for the elastic constants of crystals are not valid in a general force scheme. The expressions for the elastic constants which follow the different theoretical procedures are derived for the case of diamond and are compared with the experimental results.

      Finally, a cubic equation whose roots determine the limiting group velocities of the long acoustic waves travelling in any direction of the crystal, has been derived; this replaces the expression for the velocities of the long acoustic waves given in an earlier paper by the author.

    • The theory of the propagation of light in polycrystalline media

      C V Raman K S Viswanathan

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      A formula based on wave-theoretical considerations is deduced which gives the coefficient of extinction of plane-polarised light traversing a polycrystalline aggregate in terms of the wave-length of the light, the size of the particles and their birefringence. The general formula covers the case where the particles have preferred orientation expressible by three different probability numbers for three mutually perpendicular directions, and the special case of isotropic orientation is readily derivable therefrom. The significance of the results is discussed in relation to the facts of observation.

    • A generalised theory of the christiansen experiment

      C V Raman K S Viswanathan

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      A formula is derived for the transmission coefficient of a Christiansen cell containing particles of a birefringent material whose interstices are filled up by a liquid of suitably adjusted refractive index. The consequences of the formula and especially the influence of the birefringence on the spectral character of the transmitted light are discussed.

    • The theory of the elasticity of crystals

      K S Viswanathan

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    • The elastic behaviour of isotropic solids

      C V Raman K S Viswanathan

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    • On the theory of the elasticity of crystals

      C V Raman K S Viswanathan

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      The fundamental aspects of the phenomenological theory of elasticity are critically examined and it is shown that the tensor representation of the elastic strains and stresses in the general case should be in the unsymmetrical form. On this basis, the stress-strain relationships are deduced and tabulated for the different crystal classes. The equations determining the velocities of wave-propagation in different directions are also obtained and tabulated. Static deformation problems are then discussed and it is shown that in the particular case of homogeneous strains, the elastic constants group themselves in linear combinations which are equivalent to the elastic modulii of the theory in its familiar form. In wave-propagation, however, the strains and stresses are heterogeneous and hence all the elastic constants are involved and appear in linear combinations which are different and also larger in number than those which figure in the formulæ for homogeneous deformations. These results are completely in accord with the consequences of the atomistic theory based on interatomic forces of the most general type.

    • Corrigenda

      C V Raman K S Viswanathan

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    • The theory of the anharmonic oscillator

      K S Viswanathan

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      Exact expressions have been obtained for the frequency of an anharmonic oscillator, the shift in its equilibrium position from the origin and for the amplitudes of its different harmonics. It is shown that the frequency of the oscillator is a decreasing function of the energy and that some of the results of wave-mechanics can be obtained from the classical theory by substituting in the classical energy of the oscillator the different energy values of a harmonic oscillator. The eigenvalues of the oscillator are determined by using the W.K.B. method. The classical theory of a Morse oscillator has similarly been worked out in Appendix II.

    • Anharmonicity of vibration in molecules

      K S Viswanathan

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      The presence of anharmonicity entails the interactions of the normal modes of vibrations, which are independent in the harmonic oscillator approximation. The method of Hartree has been applied to study the mutual interaction of the normal modes, each assumed to be moving in the average potential field of the rest, and to evaluate their wave functions and eigenvalues. It is shown that, to a first order of approximation, normal vibrations belonging to the antisymmetric species do not interact with the rest and suffer no anharmonicity at all. The wave functions and eigenvalues of the different normal modes have been evaluated correct to the second order. The question of degeneracy has been considered and expressions have been given for the energy values of the different sublevels into which an overtone level of a degenerate system may be expected to split up according to group-theoretical considerations.

    • The relativistic theory of chemical binding

      K S Viswanathan

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      Starting from Breit’s relativistic equation for a system of two electrons, it is shown that for a hydrogen molecule (or for a system of two electrons moving in a field of cylindrical symmetry) the component of the total angular momentum (Jx) along the axis of the molecule (axis of symmetry) is a constant of motion. Thus every eigenstate of the system is simultaneously an eigenstate of Jx also, and a state of the system will specify, besides its energy, only the eigenvalue of the component of the angular momentum parallel to the axis of symmetry. The form of the four large components of the wave function relating to their dependence on the azimuthal co-ordinates has been given.

      The case of Russel-Saunders approximation has been considered in detail and the nature of the components of the wave function for the singlet and triplet states has been discussed. It is shown that the wave function for the ground state of the hydrogen molecule could be expressed as a sum of a set of symmetric functions of which the first term is the Heitler-London function, and that the wave function for a triplet state should be a superposition of anti-symmetric molecular orbitals. It is shown that relativistic theory brings about in a natural manner the facts relating to the ground state of the molecules C2 and O2. Finally, some remarks are made concerning the case of molecules for which the spinorbit and the spin-spin couplings are strong.

    • The Dirac equation for many-electron systems

      K S Viswanathan

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      It is shown that the Dirac equation for an electron moving in a field can be derived by expressing the magnitude of the momentum four-vector in terms of its components. By considering a four-vector whose components denote the total momentum and energy of the particles, a relativistic equation for a system of several electrons has been derived. A representation of this equation has been made in the product space of the electrons and it is shown that for the special case of a system containing two electrons, it leads to the well-known Breit equation.

    • The correlated Hartree-Fock equations and the generalised density matrices

      K S Viswanathan

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      The paper deals with a study of correlation effects in many-electron systems. Coulomb correlation is introduced into the theory by multiplying the Slater determinant formed from the one-electron orbitals by a correlation factor which is a symmetric and increasing function of the inter-electronic distances. The integro-differential equations satisfied by the best one-electron orbitals have been been deduced for non-stationary systems. From the extended Hartree-Fock equations given by Löwdin, the integro-differential equations satisfied by the density matrices have been derived. An expression for the energy-matrix of the system which is helpful in deriving a correlated Thomas-Fermi charge distribution, has also been given.

    • The many electron wave equation in the Schrödinger-Pauli form

      K S Viswanathan

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      The paper deals with the reduction of the generalised Dirac equation for a system containing N electrons to its 2N large components. The wave equation for many electron systems has been derived in its Schrödinger-Pauli form, and this includes higher order relativistic effects such as the mass change of the particles with their velocities, the spin-orbit, the orbit-orbit and the spin-spin interactions. A few remarks are made on the physical interpretation of the components of the wave function and the relations which they should obey in order to satisfy the Pauli Exclusion Principle.

    • Close equatorial satellites of the moon

      K S Viswanathan N Rajappa

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      The paper deals with the orbits of near satellites of the moon moving in its equatorial plane. The figure of the moon is a triaxial ellipsoid, and the effects of the perturbation caused by the second and fourth harmonic terms in the gravitational potential of the moon are considered. It is shown that the areal velocity of the satellite is not a constant, and its deviation from the constant value is, in the first order, proportional to the difference in the principal moments of inertia about the two axes lying in the equatorial plane. Expressions for the rotation of the major axis of the orbit are given. The equations of motion are solved forh (areal velocity) andu (reciprocal of the radial distance) correct to second-order terms in J1 and J2.

    • Turbulent shock waves in a collisionfree plasma

      K S Viswanathan G S Dwarakanath

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      The paper deals with the structure of collisionless shocks arising from turbulent wave-particle interactions. The conditions under which wave-particle interaction effects could become significant leading to growing waves and a shock are discussed. Using the Mott-Smith expression for the zero-order distribution functions for the ions within the shock, the dielectric constant as well as the integral representing the wave-particle interaction term in the Lenard-Balescu equation are evaluated for a collisionless plasma. An expression is given for the ion distribution function within the shock.

      It is shown that the component of the pressure tensor perpendicular to the direction of flow of the plasma leads to a new kind of viscosity term arising from the interaction of the particles with the growing waves and this provides a dissipative mechanism to account for the conversion of the kinetic energy of the incoming plasma into the thermal energy of the hot ionised gas behind the shock.

    • The structure of strong shock waves using the Fokker-Planck model

      K S Viswanathan G S Dwarakanath V V Shanbhag

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      The structure of strong shock waves in monatomic gases is studied using the Fokker-Planck model to represent the particle collisions and the Mott-Smith distribution to describe the distribution function within the shock front. The differential equation governing the variation of the density within the shock is derived by using the variational principle. The thickness of the shock front is evaluated numerically for various monatomic gases for Mach numbers varying from 2 to 20, and besides, the variation of the shock thickness with viscosity is also studied for different gases. Several parameters of physical interest within the shock, such as density, temperature and mean velocity of flow are evaluated numerically and detailed curves showing their variation within the shock are presented for different Mach numbers. It is found that the temperature rises very steeply, reaches a maximum within a distance less than half the thickness of the shock and then diminishes slowly to attain its asymptotic downstream values. The variation of the mean velocity is slow for weak shocks, but for higher Mach numbers, the mean velocity diminishes steeply and reaches the downstream values within half the thickness of the shock.

    • Pitch angle scattering into loss cone by cyclotron resonance

      K S Viswanathan K Revathy

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    • The propagation of ultrasonic waves in piezoelectric semiconductors

      K S Viswanathan S Rajam

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      The equations of wave propagation in piezoelectric semiconductors have been derived for a frame of reference in which the principal axes concide with the crystallographic axes. It is shown that generally the dispersion relation is given by a determinant of order six but under condition wherein the plasma modes are not excited, it could be reduced to a determinant of order five, which is equivalent to the one given by Hutson and White. The dispersion relation for hybrid waves which couple acoustic phonons with plasmons has been derived and this is shown to be given by a determinental equation of order four.

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