New Ferroelectric Ba₅Ti₂O₇Cl₄

Abstract

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.

Domain Structure in the New Ferroelectric Ba 5 Ti 2 O 7 Cl 4

“…A large inclination of 6 • shows that the displacement of the ion in this case is appreciable. This is not surprising in view of the large size of the unit cell [1] as compared to the size of the unit cell of BaTiO 3 [7] or KNbO 3 [8]. It may be mentioned here that the angular separation of such two sets of these lines has always been found to be 6 • , which shows that this is a characteristic of this type of domain lines.…”

“…Therefore, if the desired planes are also the cleavage planes, the matters become simple and advantageous. The crystal structure worked out by the authors [1] suggested that the observations of (100) or (010) and (001) planes should be advantageous, looking to the role of Ti 4+ ion in the domain formation, and the similar role of the Cl 1− ion [1]. These are also the cleavage planes for this crystal, though the cleavage is not very smooth.…”

“…It is thus seen that the angular separation of the same variety of lines is either 6 • or 12 • . One can expect more displacement for Cl 1− ion on account of its central position in the unit cell [1]. Hence, an angular separation of 6 • can be related to the Ti 4+ ion, and of 12 • to the Cl 1− ion.…”

“…One has to consider together lines of same separation on (001) plane and on (010) Polarizing microscope studies discussed in the next section show that polar axis always lies in (001) planes and has two possible inclinations that are not constant but are around 45 • . Reference to the crystal structure [1] gives that the displacement of Cl 1− ion may correspond to one polar axis, and the displacement of Ti 4+ ion to the other. Also, the geometry of the unit cell shows that the angle between the edge and the diagonal of the unit cell corresponds to the angle on average between the two varieties of domain lines on (001) plane, making allowance for the two positions of the twin plane.…”

“…Recently, a new ferroelectric Ba 5 Ti 2 O 7 Cl 4 (barium titanium oxide, barium hypochlorite) was reported from this laboratory by the authors [1]. This ferroelectric has many interesting properties, the most important amongst them being the presence of two ferroelectric mechanisms, one associated with the Ti 4+ ion, and the other with a Cl 1− ion.…”