On the Density-Density Critical Indices¶in Interacting Fermi Systems

The behaviour of correlation functions of d=1$ interacting fermionic systems is determined by a small number of critical indices. We prove that one of them is exactly zero. As a consequence, the behavior of the Fourier transform of the density-density correlation at zero momentum is qualitatively unaffected by the interaction, contrary to what happens at , if is the Fermi momentum. The result is obtained by implementing Ward identities in a Renormalization Group approach. Received: 12 November 2001 / Accepted: 25 February 2002?Published online: 2 October 2002     

Ward Identities and Vanishing of the Beta Function for d = 1 Interacting Fermi Systems

We give a self consistent and simplified proof of the (asymptotic) vanishing of the Beta function in d=1 interacting Fermi systems as a consequence of a few properties deduced from the exact solution of the Luttinger model. Moreover, since the vanishing of the Beta function is usually “proved” in the physical literature through heuristic arguments based on Ward identities, we briefly discuss here also the possibility of exploiting this idea in a rigorous approach, by using a suitable Dyson equation. We show that there are serious difficulties, related to the presence of corrections (for which we get careful bounds), which are usually neglected.     

Existence of 3D-Vortices in Abelian Gauge Theories

We consider the Abelian gauge theory related to the interaction between the solitary waves of the nonlinear Klein-Gordon equation and the electromagnetic field. We analyze the existence of electromagnetostatic solutions and, in particular, of vortices. Dedicated to the memory of Professor Aldo Cossu     

Numerosities of labelled sets: a new way of counting

The notions of “labelled set” and “numerosity” are introduced to generalize the counting process of finite sets. The resulting numbers, called numerosities, are then used to develop nonstandard analysis. The existence of a numerosity function is equivalent to the existence of a selective ultrafilter, hence it is independent of the axioms of ZFC.     

A purely algebraic characterization of the hyperreal numbers

The hyperreal numbers of nonstandard analysis are characterized in purely algebraic terms as homomorphic images of a suitable class of rings of functions.

     

Intermolecular proton transfer in solid phase: a rare example of crystal-to-crystal transformation from hydroxo- to oxo-bridged iron(III) molecule-based magnet

Intermolecular proton transfer in solid phase from the hydroxo bridge to a water molecule occurs in a new mu-hydroxo iron(III) compound of formula {EtNH3[Fe2(ox)2Cl2(mu-OH)].2H2O}n leading to a still crystalline compound in which the mu-oxo bridge replaces the mu-hydroxo one. Both three-dimensional compounds exhibit magnetic ordering at Tc ca. 70 K due to a spin canting.     

CO(2) Fixation by Novel Copper(II) and Zinc(II) Macrocyclic Complexes. A Solution and Solid State Study

Solutions containing Zn(II) and Cu(II) complexes with [15]aneN(3)O(2) rapidly adsorb atmospheric CO(2) to give {[ZnL](3)(&mgr;(3)-CO(3))}.(ClO(4))(4) (2) and {[CuL](3)(&mgr;(3)-CO(3))}.(ClO(4))(4) (4) complexes. The crystal structures of both complexes have been solved (for 2, space group R3c, a, b = 22.300(5) ?, c = 17.980(8) ?, V = 7743(4) ?(3), Z = 6, R = 0.0666, R(w)(2) = 0.1719; for 4, space group R3c, a, b = 22.292(7) ?, c = 10.096(8) ?, V = 7788(5) ?(3), Z = 6, R = 0.0598, R(w)(2) = 0.1611), and the spectromagnetic behavior of 4 has been studied. In both compounds a carbonate anion triply bridges three metal cations. Each metal is coordinated by one oxygen of the carbonate, three nitrogens, and an oxygen of the macrocycle; the latter donor weakly interacts with the metals. Although the two compounds are isomorphous, they are not isostructural, because the coordination geometries of Zn(II) in 2 and Cu(II) in 4 are different. The mixed complex {[CuZn(2)L(3)](&mgr;(3)-CO(3))}.(ClO(4))(4) has been synthesized. X-ray analysis (space group R3c, a, b = 22.323(7) ?, c = 17.989(9) ?, V = 7763(5) ?(3), Z = 6, R = 0.0477, R(w)(2) = 0.1371) and EPR measurements are in accord with a &mgr;(3)-carbonate bridging one Cu(II) and two Zn(II) ions in {[CuZn(2)L(3)](&mgr;(3)-CO(3))}(4+). Both the Zn(II) and Cu(II) cations exhibit the same coordination sphere, almost equal to that found in the trinuclear Zn(II) complex 2. The systems Zn(II)/L and Cu(II)/Lhave been studied by means of potentiometric measurements in 0.15 mol dm(-)(1) NaCl and in 0.1 mol dm(-)(3) NaClO(4) aqueous solutions; the species present in solution and their stability constants have been determined. In both systems [ML](2+) species and hydroxo complexes [M(II)LOH](+) (M = Zn, Cu) are present in solution. In the case of Cu(II), a [CuL(OH)(2)] complex is also found. The process of CO(2) fixation is due to the presence of such hydroxo-species, which can act as nucleophiles toward CO(2). In order to test the nucleophilic ability of the Zn(II) complexes, the kinetics of the promoted hydrolysis of p-nitrophenyl acetate has been studied. The [ZnLOH](+) complex promotes such a reaction, where the Zn(II)-bound OH(-) acts as a nucleophile to the carbonyl carbon. The equilibrium constants for the addition of HCO(3)(-) and CO(3)(2)(-) to the [ZnL](2+) complex have been potentiometrically determined. Only [ML(HCO(3))](+) and [ML(CO(3))] species are found in aqueous solution. A mechanism for the formation of {[ML](3)(&mgr;(3)-CO(3))}.(ClO(4))(4) is suggested.     

Synthesis and coordination properties of new macrocyclic ligands: equilibrium studies and crystal structures

The synthesis and characterization of two new macrocyclic ligands, the bis-macrocyclic compound 2,6-bis(1,4,13-triaza-7,10-dioxacyclopentadec-1-ylmethyl)phenol (L) and 38-methoxy-1,4,13,16,19,28-hexaaza-7,10,22,25-tetraoxatricyclo[14.14.7.1(32,36)]octatriconta-32,34,Delta(36,38)-triene (L1) are reported. Equilibrium studies of basicity and coordination properties toward metal ions such as Cu(II), Zn(II), Cd(II) and Pb(II) were performed for ligand by potentiometric measurements in aqueous solution (298.1 +/- 0.1 K, I= 0.15 mol dm(-3)). L behaves as a hexaprotic base (logK(1)= 10.93, logK(2)= 9.70, logK(3)= 8.79, logK(4)= 8.05, logK(5)= 6.83, logK(6)= 2.55). All metal ions form stable mono- and dinuclear complexes: logK(MLH(-1))= 25.61 for Cu(II), 15.37 for Zn(II), 12.58 for Cd(II) and 13.79 for Pb(II); logK(M(2)LH(-1))= 31.61 for Cu(II), 23.38 for Zn(II), 24.49 for Cd(II) and 23.68 for Pb(II). All these dinuclear species show a great tendency to add the OH(-) group: the equilibrium constant for the addition reaction was found to be logK(M(2)LH(-1)OH)= 4.77 for Cu(II), 5.66 for Zn(II), 2.8 for Cd(II) and 3.18 for Pb(II). In the case of Ni(II), kinetic inertness prevents the possibility of solution studies. The dinuclear solid adducts [Ni(2)H(-1)L(N(3))(3)].EtOH and [Cu(2)H(-1)L(N(3))](ClO(4))(2) were characterized by X-ray analysis.     

Selective Detection of Mg2+ for Sensing Applications in Drinking Water

A new series of ligands containing the 2-(2-hydroxy-3- naphthyl)-4-methylbenzoxazole (HNBO) fluorophore showed selectivity for Mg²⁺ ions, without the interference of Ca²⁺. The most promising representative L3 resulted the best performing sensor for Mg²⁺ both in solution and embedded in an all-solid-state optode, especially towards real samples of drinkable water.