Interferometric studies of domain structures in KNbO3 single crystals, grown by using excess of K2CO3 as a flux are reported. The surface deformations produced on the pseudocubic (001) planes by the stable patterns of 90°, 60° and 180° domains have been studied and discussed. If the observed plane is not the plane of the shear, the 90° domains are inclined at 30prime. The angle between the 60° domains is 57prime for the domain line, parallel to a pseudocubic edge. The head to tail coupling of dipoles across the 60° domain walls gives a step ladder structure on the surface. The nature of dipole coupling in a step ladder due to 90° domains cannot, however, be distinguished. The wedges are all 60° wedges, with the wedge angle 57prime. The theoretical formulae have been derived, and the values found to agree closely with the experimental values.The angles between the planes and the other surface angles are all temperature dependent as they are functions of the lattice parameters. In the tetragonal phase, the domain behaviour is found to be quite similar to that of BaTiO3. The domain patterns are seen at times to have a memory indicating the presence of traps in the crystals for these domain walls.

It has been shown theoretically as well as experimentally that microwalls with linear dimensions of the order of 10−6–10−8 m can be nucleated in KNbO3 single crystals under the influence of the electric dc fields applied externally at the sites of the impurity ions. The critical field for nucleation is found to be about 8×103 V m−1. Since such low fields already exist in crystals at the nonferroelectric to ferroelectric phase transition, the mechanism is expected to be operative at that time, and a photographic evidence of it is also obtained. A model has been developed to explain the nucleation. Employing thermodynamic considerations, expressions have been derived for the critical length of the domain wall nucleated, and the activation energy of nucleation, both for 60° and 90° walls. The photomicrographs obtained after applying the dc fields show microwalls nucleated at the impurity sites, with their linear dimensions of the order given by the theory. Many other aspects of the domain wall nucleation have been made clear by the photomicrographs. Particularly, it has been found that the microwalls move after nucleation to lie linearly to reduce the strain energy of the crystal. The microwalls are not pinned at the impurities, and extend themselves so that they meet each other end to end, producing a continuous wall. It is possible to distinquish these microwalls from the microwalls nucleated by the dislocation loops. The expression for the activation energy of nucleation is also verified experimentally. It is suggested that the mechanism of domain wall nucleation can occur in other ferroelectric crystals also, and shown that the memory of domain walls, commonly observed in ferroelectrics, can be readily attributed to the impurity ions present in the crystals, and the phenomenon of domain wall nucleation operative at their sites.

The non-commutative O(N) Gross-Neveu model is solved in the large N limit in two and three space-time dimensions. The commutative version of the two dimensional model is a renormalizable quantum field theory, both in a coupling constant expansion and an expansion in 1/N . The non-commutative version has a renormalizable coupling constant expansion where ultraviolet divergences can be removed by adjusting counterterms to each order. On the other hand, in a previous work [1] we showed that the non-commutative theory is not renormalizable in the large N expansion. This is argued to be due to a combined effect of asymptotic freedom and the ultraviolet/infrared mixing that occurs in a non-commutative field theory. In the present paper we will elaborate on this result and extend it to study the large N limit of the three dimensional Gross-Neveu model. We shall see that the large N limit of the three dimensional theory is also trivial when the ultraviolet cutoff is removed.

Switching studies on KNbO3 single crystals were carried out under DC electric field conditions. The single crystals, both the major faces of which were photographed to note down the details of domain structure, were subjected to steady electric fields of known value and duration. The disturbances produced in the domain structure form part of the switching process. These changes were studied by photographing the faces immediately after the field was removed. It is found that switching proceeds through the nucleation of domain walls rather than the nucleation of fresh domains, and the lateral and longitudinal movement of domain walls. There exists a critical field of about 80 V cm-1 for nucleation. The walls are usually nucleated in pairs or in even numbers. The number of walls so nucleated for a given field applied for a given time depends upon the defect conditions of the crystal. As regards wall movements, it is found that the lateral wall velocities are comparable with forward/backward wall velocities. The 90 degrees and 60 degrees walls can be distinguished by their different way of movement. Thus the method provides a direct probe into the switching mechanism, revealing in the present case an important step in switching, viz. the nucleation of domain walls.

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