Solutions to the nonlinear Schrödinger equation with sequences of initial data converging to a Dirac mass

We study the nonlinear Schrödinger equation with sequences of initial data that converge to a Dirac mass, and study the asymptotic behaviour of solutions. In doing so we find a connection to previously known long time asymptotics. We demonstrate a type of universality in the behaviour of solutions for real initial data, and we also show how this universality breaks down for examples of initial data that are not purely real.     

Crystal Structures and Magnetic Properties of Rare-Earth Ultraphosphates, RP5O14 (R=La, Nd, Sm, Eu, Gd)

The single-crystal X-ray structures of lanthanum, europium, and gadolinium ultraphosphate, RP₅O₁₄ (R=rare-earth) are reported herein [monoclinic, P2₁/c, a=8.8206(1), 8.7491(1), 8.7493(1) Å, b=9.1196(2), 8.9327(1), 8.9189(1) Å, c= 13.1714(2), 12.9768(2), 12.9717(1) Å, β=90.661(1), 90.534(1), 90.6682(3)°, respectively; Z=4; R1=0.0250, 0.0346, 0.0270, respectively]. The structures are all type (I) compounds as classified by Bagieu-Beucher and Tranqui [Bull. Soc. Fr. Miner. Cryst. 93, 505 (1970)]. The minimum R…R separations are compared with all other structural reports of lanthanide ultraphosphates. Type (I) compounds have the lowest minimum R…R separation, which decreases with atomic number and appears not to perturb the optical properties of any rare-earth ultraphosphate. In each case, R is surrounded exclusively by eight oxygen atoms that form a distorted square antiprism. A P–O network holds together the three-dimensional structure. The magnetic susceptibilities of neodymium, samarium, and gadolinium ultraphosphate as a function of temperature are also reported along with corresponding magnetization measurements. All compounds exhibit a paramagnetic response, following Curie's law except in the regions where crystal field splittings are significant.     

Comparing the Atomic Structures of Binary MO2-SiO2 (M = Ti, Zr or Hf) Xerogels

The incorporation of transition-metal oxides into silica can give materials with useful optical, electronic or catalytic properties. For example, ZrO₂-SiO₂ and HfO₂-SiO₂ materials are of interest due to their high dielectric constants. Here we present a comparison of extended X-ray absorption fine structure and small-angle X-ray scattering results for acid-catalysed binary (MO₂) _x (SiO₂)_{1 – x} (M = Ti, Zr or Hf) xerogels, with x up to 0.4 and heat treatments up to 750°C. Detailed observations for TiO₂-SiO₂ and ZrO₂-SiO₂ xerogels provide a basis for interpretation of new results for HfO₂-SiO₂ xerogels. At low concentrations metal atoms are homogeneously incorporated into the silica network. Ti adopts coordinations of 4 or 6, and Zr and Hf both adopt higher coordination of 6 or 7 (the larger coordinations being due to ambient moisture). At higher concentrations, phase separation of metal oxide occurs. Such regions become clearly separated from the silica network for TiO₂, but remain very finely mixed with silica network for ZrO₂ and HfO₂.     

Characterisation of sol-gel prepared (HfO2)x(SiO2)1-x (x=0.1, 0.2 and 0.4) by 1H, 13C, 17 O and 29Si MAS NMR, FTIR and TGA

The HfO2-SiO2 system is attracting interest as a possible new dielectric material in semiconductor devices. Knowledge of the location of hafnium within the silica network and the effect hafnium has on the structure will be central to the successful use of this material system in this application. Here, sol-gel techniques have been used to manufacture (HfO2)x(SiO2)1-x samples (x=0.1, 0.2 and 0.4, each heat treated at 250, 500 and 750 degrees C) and these have been characterised by magic angle spinning (MAS) NMR (1H, 13C, 17 O, 29Si), Fourier transform infrared spectroscopy (FTIR) and thermogravimetric analysis. 29Si MAS NMR showed that increasing the hafnia content decreases the connectivity of the silicate network, i.e. increases the range of differently connected SiO4 (Qn) units with more having increased numbers of non-bridging oxygens (i.e. lower n). FTIR and 17 O MAS NMR showed unequivocally that the x=0.4 sample phase-separated at higher temperatures, while in the x=0.1 sample the hafnium was homogeneously mixed into the SiO2 phase without any phase separation.     

Transition Metal Atom Sites in Ternary ZrO2-TiO2-SiO2 Xerogels

There has been much work on the binary TiO₂-SiO₂ and ZrO₂-SiO₂ materials prepared by sol-gel because of the beneficial properties resulting from incorporation of Ti and Zr. In contrast the ternary TiO₂-ZrO₂-SiO₂ xerogels have been relatively little studied. We report the results of a study of those xerogels having Zr:Ti:Si ratios of 5:15:80, 10:10:80 and 15:5:80 heated to 750°C and to 1000°C. The study includes X-ray diffraction, small angle X-ray scattering, X-ray absorption spectroscopy at Ti and Zr K-edges, and ¹⁷O MAS-NMR. The study has benefited from close comparison with similar previous studies of the binary systems. The metal atoms in the ternary systems are shown to be predominantly homogeneously mixed in the silica network, as observed for the respective binary systems. The clear exception is for the sample with a minority of Zr, which after heat treatment at 750°C shows the presence of phase separation attributed to the formation of an amorphous precursor of ZrTiO₄; at 1000°C this phase crystallises. In samples with higher Zr content the crystallisation of a ZrO₂ tetragonal phase was observed. The data obtained illustrate well the strength of a research methodology in which a common batch of samples is studied using a coherent suite of modern structural probes.     

Synthesis of titanium(IV) guanidinate complexes and the formation of titanium carbonitride via low-pressure chemical vapor deposition

The mono(guanidinato) complex [Ti(NMe2)2Cl{i-PrNC[N(SiMe3)2]N-i-Pr}] (1) was prepared by reaction of [Ti(NMe2)2Cl2] with 1 or 2 equiv of the lithium guanidinate salt [Li{i-PrNC[N(SiMe3)2]N-i-Pr}]. Compound 1 has been characterized by X-ray crystallography. Treatment of TiCl4 with 2 equiv of [Li{i-PrNC[N(SiMe3)2]N-i-Pr}] resulted in the formation of dark red crystals. X-ray crystallography showed that these crystals consist of a 70:30 mixture of two bis(guanidinato) complexes, namely, [TiCl2{i-PrNC[N(SiMe3)2]N-i-Pr}{i-PrNC(N=CMe2)N-i-Pr}] (2) and [TiCl2{i-PrNC[N(SiMe3)2]N-i-Pr}{i-PrNC[N(H)-i-Pr]N-i-Pr}] (3). Both compounds 2 and 3 possess a transformed guanidinate ligand. Low-pressure chemical vapor deposition of either compound 1 or [TiCl2{i-PrNC(NMe2)N-i-Pr}] (4) at 600 degrees C results in thin films of titanium carbonitride.