Identifying the Sources of Ferromagnetism in Sol-Gel Synthesized Zn\u3csub\u3e1-x\u3c/sub\u3eCo\u3csub\u3ex\u3c/sub\u3eO (0 ≤ x ≤ 0.10) Nanoparticles

We have carefully investigated the structural, optical and electronic properties and related them with the magnetism of sol-gel synthesized Zn1-xCoxO (0 ≤ x ≤ 0.10) nanoparticles. Samples with x ≤ 0.05 were pure and free of spurious phases, whereas ZnCo2O4 was identified as the impurity phase for samples with x ≥ 0.08. Samples with x \u3c 0.05 were found to be true solid solutions with only high spin Co2+ ions into ZnO structure, whereas sample with x = 0.05, exhibited the presence of high spin Co2+ and low spin Co3+. For the impurity-free samples we found that as Co concentration increases, a and c lattice parameters and Zn–O bond length parallel to the c-axis decrease, the band gap drastically decreases, and the average grain size and distortion degree increases. In all samples there are isolated Co2+ ions that do not interact magnetically at room temperature, bringing about the observed paramagnetic signal, which increases with increasing Co concentration. M vs T curves suggest that some of these disordered Co2+ ions in Zn1−xCoxO are antiferromagnetically coupled. Moreover, we also found that the intensity of the main EPR peak associated to Co2+ varies with the nominal Co content in a similar manner as the saturation magnetizations and coercive fields do. These results point out that the ferromagnetism in these samples should directly be correlated with the presence of Co2+. Bound magnetic polaron model is insufficient to explain the Room temperature ferromagnetism in these Co doped ZnO samples and the charge transfer model seems not influence considerably the FM properties of Zn1-xCoxO nanoparticles. The FM behavior may be originated from a combination of several factors such as the interaction of high spin Co2+ ions, the formation of defect levels close to the valence band edge and grain boundaries effects

Integrability of stochastic birth-death processes

Stochastic birth-death processes are described as continuous-time Markov processes in models of population dynamics. A system of infinite, coupled ordinary differential equations (the so-called master equation) describes the time-dependence of the probability of each system state. Using a generating function, the master equation can be transformed into a partial differential equation. In this contribution we analyze the integrability of two types of stochastic birth-death processes (with polynomial birth and death rates) using standard differential Galois theory. We discuss the integrability of the PDE via a Laplace transform acting over the temporal variable. We show that the PDE is not integrable except for the case in which rates are linear functions of the number of individuals