S. Sengupta’s research while affiliated with Jadavpur University and other places

It is shown that starting from a Fourier transform relation one can derive, in a surprisingly simple manner, all the well-known results of lattice summation, that have been obtained so far by a complicated use of the Ewald theta transformation. We show that the Ewald transformation follows directly from the Fourier transform relation.

A self‐consistent solution for the effective elastic properties of polycrystalline and perfectly disordered multiphase composites has been discussed by using the T‐matrix method under certain suitable approximations. Compared to the existing formulas these new relations for the disordered composites are very useful in practical situations for a quick and more accurate estimate of the effective elastic properties, in particular for a case where the composite has components with widely different values of the elastic constants. For comparison we have discussed the results based on Kröner’s theory which also purports to solve the same problem. It is found that the two solutions do not agree. To resolve the difference we take help of Hill’s exact solution of the composite problem when the components have equal rigidities. It is found that while Kröner’s theory is inconsistent with the exact result the present self‐consistent solution analytically reproduces it. Another interesting finding of the present investigation is that the approximations made in obtaining the self‐consistent solution are exact in the limit of composites with equal‐shear moduli. Finally it is indicated that although the results for composites have been derived for isotropic and cubic components it can be easily adapted for a composite with noncubic components.

In order to calculate the electrostatic contributions to the elastic constants of piezoelectric crystals, Fuller and Naimon [Phys. Rev. B. 6, 3609 (1972)] used the conventional method of homogeneous deformation by evaluating the lattice sums using the Ewald transformation and omitting the zero-wave-vector term. This procedure, however, lacks justification, since it is well known that the conventional homogeneous deformation theory breaks down for piezoelectric crystals. We develop here the full theory for this case. (The use of a new technique for performing the theta transformation makes the treatment much simpler.) The theory verifies that the procedure of Fuller and Naimon will give the correct contribution to the elastic constants. We also explain why the zero-wave-vector term remained absent in their treatment. Moreover, we derive for an arbitrary crystal structure some relationships between the electrostatic contributions to the different second- and third-order elastic constants. In view of these relationships, one has to calculate a lesser number of electrostatic contributions for a given crystal structure, and some of the evaluations of Fuller and Naimon become redundant.

A simple procedure involving sound velocities in component crystallites has been developed to calculate the effective elastic properties of a hypothetical aggregate which is macroscopically homogeneous but has fluctuations on a microscopic scale, with the underlying assumption that the sound wave can distinguish between the different crystallites oriented at random. For comparison we have also employed the effective medium theory based on a static deformation scheme to calculate the effective elastic properties of the same hypothetical aggregate. Both the procedures use the identical set of input data. Although the assumptions on which the above dynamic and static methods are based are entirely different the results obtained from them for seventeen different aggregates are remarkably close to each other and also to experiments done on polycrystalline specimens using ultrasonic velocities. Finally, a critical discussion of the results and their consequences is presented.

We derive expressions for the third-order elastic, piezoelectric, and dielectric constants, in terms of the interaction potential, and show that in fluorite structure, equal and opposite dipole moments under homogeneous strain develop. The presence of such latent polarisation is reflected in the dielectric response of the strained crystal.

The lattice sum Sigma r-n cos(ar) exp(i alpha .r) has not yet been tackled for an arbitrary value of n. Here the authors obtain quite easily a rapidly convergent expression for this sum, by using a Fourier transform relationship. This result is applied to evaluate the Fourier transform of the Ruderman-Kittel interaction. It is shown that from a special case of the Fourier transform relationship, one may derive the Ewald transformation.

In the context of the general problem of equivalence between classical mechanics and quantum mechanics in the macroscopic limit, we point out that, for the particular case of the one-dimensional Coulomb potential, the quantum-mechanical result in the classical limit, corresponding to a certain superposition of odd- and even-parity energy eigenfunctions, leads to inconsistency with classical mechanics. It is shown that the contradiction persists even if the singularity of the Coulomb potential is treated as the limiting case of a modified Coulomb potential in which the singularity has been smoothed out. The possible implication of this paradoxical finding is briefly discussed.

With a view to exploring the possibility of wider implication of Bell's theorem, we argue that Bell's inequality is derivable as a general consequence of non-contextual hidden-variable theories. We formulate a new type of gedanken example indicating incompatibility of Bell's inequality with quantum mechanical predictions concerning simultaneous measurement of commuting observables associated with a system having no spatially separated components, where the locality condition is not at issue. Significance of this example is pointed out.

The specific form of the change in overlap interaction between ions due to the presence of dipole polarizability implied by the shell model lacks any microscopic justification, while all other terms of the model have a sound quantum-mechanical basis. Recently we have shown that within the framework of the Heitler-London approach this modified overlap term of the shell model is microscopically justifiable. This calculation provides a method of evaluating the parameters of the model directly from the wave functions of the ions. Since the shell model has been successful in describing the different lattice-mechanical properties, it seems instructive to investigate how far it can reproduce the collective dynamics of the electrons in insulators. Starting from the Hartree-Fock wave function of the ions the calculation is presented for the dispersion relations of the collective oscillation of the shells in the shell model by relaxing the usual adiabatic condition for the two crystals: namely, KCl and NaCl. It is interesting to note that the q⃗→0 frequencies of these collective oscillations of shells of these two crystals together with those of the other fourteen crystals obtained phenomenologically are found to compare satisfactorily with the measured plasma frequencies.