Convex lattice polygons of fixed area with perimeter dependent weights

We study fully convex polygons with a given area, and variable perimeter length on square and hexagonal lattices. We attach a weight t^m to a convex polygon of perimeter m and show that the sum of weights of all polygons with a fixed area s varies as s^{-theta_{conv}} exp[K s^(1/2)] for large s and t less than a critical threshold t_c, where K is a t-dependent constant, and theta_{conv} is a critical exponent which does not change with t. We find theta_{conv} is 1/4 for the square lattice, but -1/4 for the hexagonal lattice. The reason for this unexpected non-universality of theta_{conv} is traced to existence of sharp corners in the asymptotic shape of these polygons.Comment: 8 pages, 5 figures, revtex

Branched Polymers on the Given-Mandelbrot family of fractals

We study the average number A_n per site of the number of different configurations of a branched polymer of n bonds on the Given-Mandelbrot family of fractals using exact real-space renormalization. Different members of the family are characterized by an integer parameter b, b > 1. The fractal dimension varies from $log_{_2} 3$ to 2 as b is varied from 2 to infinity. We find that for all b > 2, A_n varies as $\lambda^n exp(b n ^{\psi})$, where $\lambda$ and $b$ are some constants, and $0 < \psi <1$. We determine the exponent $\psi$, and the size exponent $\nu$ (average diameter of polymer varies as $n^\nu$), exactly for all b > 2. This generalizes the earlier results of Knezevic and Vannimenus for b = 3 [Phys. Rev {\bf B 35} (1987) 4988].Comment: 24 pages, 8 figure

Absence of ferromagnetism in Mn- and Co-doped ZnO

Following the theoretical predictions of ferromagnetism in Mn- and Co-doped ZnO, several workers reported ferromagnetism in thin films as well as in bulk samples of these materials. While some observe room-temperature ferromagnetism, others find magnetization at low temperatures. Some of the reports, however, cast considerable doubt on the magnetism of Mn- and Co-doped ZnO. In order to conclusively establish the properties of Mn- and Co-doped ZnO, samples with 6 percent and 2 percent dopant concentrations, have been prepared by the low-temperature decomposition of acetate solid solutions. The samples have been characterized by x-ray diffraction, EDAX and spectroscopic methods to ensure that the dopants are substitutional. All the Mn- and Co-doped ZnO samples (prepared at 400 deg C and 500 deg C) fail to show ferromagnetism. Instead, their magnetic properties are best described by a Curie-Weiss type behavior. It appears unlikely that these materials would be useful for spintronics, unless additional carriers are introduced by some means.Comment: 23 pages, 9 figures. submitted to J. Mater. Chem 200

Bethe approximation for a system of hard rigid rods: the random locally tree-like layered lattice

We study the Bethe approximation for a system of long rigid rods of fixed length k, with only excluded volume interaction. For large enough k, this system undergoes an isotropic-nematic phase transition as a function of density of the rods. The Bethe lattice, which is conventionally used to derive the self-consistent equations in the Bethe approximation, is not suitable for studying the hard-rods system, as it does not allow a dense packing of rods. We define a new lattice, called the random locally tree-like layered lattice, which allows a dense packing of rods, and for which the approximation is exact. We find that for a 4-coordinated lattice, k-mers with k>=4 undergo a continuous phase transition. For even coordination number q>=6, the transition exists only for k >= k_{min}(q), and is first order.Comment: 10 pages, 10 figure