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We prove surface and volume mean value formulas for classical solutions to uniformly parabolic equations in divergence form. We then use them to prove the parabolic strong maximum principle and the parabolic Harnack inequality. We emphasize that our results only rely on the classical theory, and our arguments follow the lines used in the original theory of harmonic functions. We provide two proofs relying on two different formulations of the divergence theorem, one stated for sets with almost C^1 boundary, the other stated for sets with finite perimeter., Comment: 33 pages, 6 figures

Significant insights for critical current density (Jc) improvement in Bi-2212 super- conductor wires can be obtained by an accurate analysis of the structural and mi- crostructural properties evolving during the so-called partial-melt process, a heat treatment needed to improve grain connectivity and therefore gain high Jc. Here, we report an in-situ analysis by means of synchrotron X-ray and neutron diffraction performed, for the first time, during the heat treatment carried out with the very same temperature profile and reacting oxygen atmosphere in which the Bi-2212 wires are usually treated for practical applications. The obtained results show the thermal evolution of the Bi-2212 structure, focusing in particular on texturing and secondary phases formation. The role of the oxygen is discussed as well. Hence, the present investigation marks a significant advance for the comprehension of the phenomena involved in the wire fabrication process and provides useful insights for the process optimization as well., Comment: to be published on Physical Review Materials

In this work we investigate the effect of each different heat treatment stage in the fabrication of Bi2Sr2CaCu2O8+d superconducting wires on intra-grain and inter-grain superconducting properties. We measure magnetic critical temperature Tc values and transport critical current density Jc at temperatures from 4 K to 40 K and in fields up to 7 T. From an analysis of the temperature dependence of the self-field critical current density Jc(T) that takes into account weak link behavior and proximity effect, we study the grain boundaries (GB) transparency to supercurrents and we establish a relationship between GB oxygenation in the different steps of the fabrication process and the GB transparency to supercurrents. We find that grain boundary oxygenation starts in the first crystallization stage, but it becomes complete in the plateau at 836 {\deg}C and in slow cooling stages, and is further enhanced in the prolonged post annealing step. Such oxygenation makes GBs more conducting, thus improving the inter-grain Jc value and temperature dependence. On the other hand, from the inspection of the Tc values in the framework of the phase diagram dome, we find that grains are oxygenated already in the crystallization step up to the optimal doping, while successive slow cooling and post annealing treatments further enhance the degree of overdoping, especially if carried out in oxygen atmosphere rather than in air.

We examine the polynomial analogues of McMullen's and Zaremba's conjectures on continued fractions with bounded partial quotients. It has already been proved by Blackburn that if the base field is infinite, then the polynomial analogue of Zaremba's conjecture holds; we will prove this again with a different method and examine some known results for finite base fields. Translating to the polynomial setting a result of Mercat, we will prove that the polynomial analogue of McMullen's conjecture holds over infinite algebraic extensions of finite fields and that, over finite fields, it would be a consequence of the polynomial analogue of Zaremba's conjecture. We will then prove that the polynomial analogue of McMullen's conjecture holds over uncountable base fields, over $\overline{\mathbb Q}$ (thanks to the theory of reduction of a formal Laurent series modulo a prime) and over number fields. For this purpose, we will examine the connection between the continued fractions of polynomial multiples of $\sqrt D$ and pullbacks of generalized Jacobians of the hyperelliptic curve $U^2=D(T)$., Comment: Corrected typos in Chapter 3

We propose a laboratory experience aimed at undergraduate physics students to understand the main features of the photoelectric effect and to perform a measurement of the ratio h/e, where h is the Planck's constant and e is the electron charge. The experience is based on the method developed by Millikan for his measurements on the photoelectric effect in the years from 1912 to 1915. The experimental setup consists of a photomultiplier tube (PMT) equipped with a voltage divider properly modified to set variable retarding potentials between the photocathode and the first dynode, and a set of LEDs emitting at different wavelengths. The photocathode is illuminated with the various LEDs and, for each wavelength of the incident light, the output anode current is measured as a function of the retarding potential applied between the cathode and the first dynode. From each measurement, a value of the stopping potential for the anode current is derived. Finally, the stopping potentials are plotted as a function of the frequency of the incident light, and a linear fit is performed. The slope and the intercept of the line allow respectively to evaluate the ratio h/e and the ratio W/e, where W is the work function of the photocathode., Comment: 14 pages, 4 figures

The first step towards high critical currents in Bi-2212 wires was the comprehension that the supercurrent is blocked over long lengths by filament-diameter bubbles grown during the melt stage, which cause expansion of the wire diameter and dedensification of the superconducting filaments. Whereas the previous successful approach to reduce the problem of voids related to bubbles was based on the application of a high overpressure during the heat treatment, we fabricated Bi-2212 wires by applying a new concept of suitably alternating groove-rolling and drawing techniques with the aim of densifying the phase already during the working procedure prior to the heat treatment. We here for the first time were able to reach in wires reacted with closed ends - i.e. with gas trapped in the wire as it happens in long-length wires - the very same values of critical current shown in short wires reacted with open ends. This is the irrefutable evidence that, only by acting on the deformation technique, we were able to raise the critical current by properly densifying the superconducting powder inside the filaments already before the melt stage. Whole-conductor current densities in our long length simulation wires already reach 400 A/mm2 at 4.2 K and 5 T, which can be still easily increased through architecture optimization. The actual breakthrough is that the densification is optimized without further complex treatments through a technique which can be straightforwardly applied to long-lengths wires., Comment: 9 pages, 3 figures

Among the recently discovered Fe-based superconducting compounds, the (K,Ba)Fe2As2 phase is attracting large interest within the scientific community interested in conductor developments. In fact, after some years of development, critical current densities Jc of about 105 A/cm2 at fields up to more than 10 T have been obtained in powder in tube (PIT) processed wires and tapes. Here we explore the crucial points in the wire/tape fabrication by means of the ex-situ PIT method. We focus on scaling up processes which are crucial for the industrial fabrication. We analyzed the effects on the microstructure of the different heat and mechanical treatments. By an extensive microstructural analysis correlated with the transport properties we addressed the issues concerning the phase purity, the internal porosity and crack formation in the superconducting core region. Our best conductors with a filling factor of about 30 heat treated at 800 C exhibited Tc = 38 K the highest value measured in such kind of superconducting tape. The microstructure analysis shows clean and well connected grain boundaries but rather poor density: The measured Jc of about 3 x 10^4 A/cm2 in self-field is suppressed by less than a factor 7 at 7 T. Such not yet optimized Jc values can be accounted for by the reduced density while the moderate in-field suppression and a rather high n-factor confirm the high homogeneity and uniformity of these tapes.

In this work, the feasibility condition of Powder-In-Tube (PIT) processed wires of Fe(Se,Te) superconductor has been investigated. We faced several technical issues that are extensively described and discussed. In particular, we tested different metals and alloys as external sheaths (Cu, Ag, Nb, Ta, Ni, Fe, cupronickel, brass) concluding that the only sheath that does not affect substantially the Fe(Se,Te) phase is Fe. On the other hand, Fe sheath introduces excess iron in the Fe(Se,Te) phase, which affects the superconducting properties; we investigated the effects of the thermal treatments and of the powder composition in order to avoid it. The maximum Jc value obtained in our samples is 4*10^2 A/cm2, comparable to other published values of PIT conductors of the 11 family. We conclude that the fabrication of Fe(Se,Te) wires by PIT method is quite challenging and other approaches should be developed.

In the present paper we report an in-situ high-energy X-ray diffraction analysis of MgB2 tapes during the preparation process. The experiment was performed in a specifically designed furnace working in reducing atmosphere, compatible with the Laue diffraction condition. The MgB2 synthesis was realized starting from MgH2 and amorphous B in powder form as precursors, varying reaction temperature and testing different cooling processes. We analyzed both the MgB2 synthesis and the sintering process of tapes prepared with these powders. Phase evolution, micro and crystallographic structure were monitored during the different thermal treatments. Among the main results we observed the formation of MgB2 at an extraordinary low temperature (300C), probably as a result of a solid-state reaction between MgH2 and B. Furthermore, we studied the dependence of the micro-structure upon the thermal treatment and its effect on the critical current performance of the superconducting tapes., Comment: 15 pages, 9 figures

Bi2Sr2CaCu2O8+x (Bi-2212) superconducting long-length wires are mainly limited in obtaining high critical currents densities (JC) by the internal gas pressure generated during the heat treatment, which expands the wire diameter and dedensifies the superconducting filaments. Several ways have been developed to increase the density of the superconducting filaments and therefore decreasing the bubble density: much higher critical currents have been reached always acting on the final as-drawn wires. We here try to pursue the same goal of having a denser wire by acting on the deformation technique, through a partial use of the groove-rolling at different wire processing stages. Such technique has a larger powders compaction power, is straightforwardly adaptable to long length samples, and allows the fabrication of samples with round, square or rectangular shape depending on the application requirements. In this paper we demonstrate the capability of this technique to increase the density in Bi-2212 wires which leads to a three-fold increase in Jc with respect to drawn wires, making this approach very promising for fabricating Bi-2212 wires for high magnetic field magnets, i.e. above 25 T.

It is well known that longer Bi-2212 conductors have significantly lower critical current density (Jc) than shorter ones, and recently it has become clear that a major cause of this reduction is internal gas pressure generated during heat treatment, which expands the wire diameter and dedensifies the Bi-2212 filaments. Here we report on the length-dependent expansion of 5 to 240 cm lengths of state-of-the-art, commercial Ag alloy-sheathed Bi-2212 wire after full and some partial heat treatments. Detailed image analysis along the wire length shows that the wire diameter increases with distance from the ends, longer samples often showing evident damage and leaks provoked by the internal gas pressure. Comparison of heat treatments carried out just below the melting point and with the usual melt process makes it clear that melting is crucial to developing high internal pressure. The decay of Jc away from the ends is directly correlated to the local wire diameter increase, which decreases the local Bi-2212 filament mass density and lowers Jc, often by well over 50%. It is clear that control of the internal gas pressure is crucial to attaining the full Jc of these very promising round wires and that the very variable properties of Bi-2212 wires are due to the fact that this internal gas pressure has so far not been well controlled.

Recently the interest about Bi-2212 round wire superconductor for high magnetic field use has been enhancing despite the fact that an increase of the critical current is still needed to boost its successful use in such applications. Recent studies have demonstrated that the main obstacle to current flow, especially in long wires, is the residual porosity inside these Powder-In-Tube processed conductors which develops in bubbles-agglomeration when the Bi-2212 melts. Through this work we tried to overcome this issue acting on the wire densification by changing the deformation process. Here we show the effects of groove-rolling versus drawing process on the critical current density JC and on the microstructure. In particular, groove-rolled multifilamentary wires show a JC increased by a factor of about 3 with respect to drawn wires prepared with the same Bi-2212 powder and architecture. We think that this approach in the deformation process is able to produce the required improvements both because the superconducting properties are enhanced and because it makes the fabrication process faster and cheaper.

The MgB2 superconductor has already demonstrated its applicative potential, in particular for DC applications such as MRI magnets, thanks to the low costs of the raw materials and to its simple production process. However further efforts have still to be made in order to broaden its employment also towards the AC applications such as SFCL, motors, transformers. The main issues are related to the reduction of the AC losses. Some of these can be faced by obtaining multifilamentary conductors with a large number of very fine filaments and, in this context, the powders granulometry can play a crucial role. We have prepared MgB2 starting powders with different granulometries and by the ex-situ P.I.T method we have realized multifilamentary wires with a number of filaments up to 361 and an average size of each filament lowered down to 30 microns. In particular we have studied the relationship between grain and filament size in terms of transport properties and show that the optimization of this ratio is possible in order to obtain suitable conductors for AC industrial applications.

We measure magnetotransport of F doped SmFeAsO samples up to 28T and we extract the upper critical fields, using different criteria. In order to circumvent the problem of criterion-dependence Hc2 values, we suggest a thermodynamic estimation of the upper critical field slope dHc2/dT based on the analysis of conductivity fluctuations in the critical regime. A high field slope as large as -12T/K is thus extracted for the optimally doped sample. We find evidence of a two-dimensional lowest Landau level (LLL) scaling for applied fields larger than mu_0H_LLL=8T. Finally, we estimate the coherence length values and we observe that they progressively increase with decreasing Tc. In all cases, the coherence length values along the c axis are smaller than the interplanar distance, confirming the two-dimensional nature of superconductivity in this compound., Comment: 14 pages, 4 figures, 3 tables Submitted to Phys.Rev.B

MgB2 monofilamentary nickel-sheated tapes and wires were fabricated by means of the ex-situ powder-in-tube method using either high-energy ball milled and low temperature synthesized powders. All sample were sintered at 920 C in Ar flow. The milling time and the revolution speed were tuned in order to maximize the critical current density in field (Jc): the maximum Jc value of 6 x 10e4 A/cm2 at 5 K and 4 T was obtained corresponding to the tape prepared with powders milled for 144h at 180rpm. Vorious synthesis temperature were also investigated (730-900 C) finding a best Jc value for the wire prepared with powders synthesized at 745 C. We speculate that this optimal temperature is due to the fluidifying effect of unreacted magnesium content before the sintering process which could better connect the grains.

Ex-situ Powder-In-Tube MgB2 tapes prepared with ball-milled, undoped powders showed a strong enhancement of the irreversibility field H*, the upper critical field Hc2 and the critical current density Jc(H) together with the suppression of the anisotropy of all of these quantities. Jc reached 104 A/cm2 at 4.2 K and 10 T, with an irreversibility field of about 14 T at 4.2 K, and Hc2 of 9 T at 25 K, high values for not-doped MgB2. The enhanced Jc and H* values are associated with significant grain refinement produced by milling of the MgB2 powder, which enhances grain boundary pinning, although at the same time also reducing the connectivity from about 12% to 8%. Although enhanced pinning and diminished connectivity are in opposition, the overall influence of ball milling on Jc is positive because the increased density of grains with a size comparable with the mean free path produces strong electron scattering that substantially increases Hc2, especially Hc2 perpendicular to the Mg and B planes., Comment: 26 pages, 9 figures, submitted to J. Appl. Phys

In DC and AC practical applications of MgB2 superconducting wires an important role is represented by the material sheath which has to provide, among other things, a suitable electrical and thermal stabilization. A way to obtain a large enough amount of low resistivity material in to the conductor architecture is to use it as external sheath. In this paper we study ex-situ multifilamentary MgB2 wires using oxide-dispersion-strengthened copper (GlidCop) as external sheath in order to reach a good compromise between critical current density and thermal properties. We prepared three GlidCop samples differing by the content of dispersed sub-microscopic Al2O3 particles. We characterized the superconducting and thermal properties and we showed that the good thermal conductivity together the good mechanical properties and a reasonable critical current density make of GlidCop composite wire a useful conductor for applications where high thermal conductivity is request at temperature above 30K, such as Superconducting-FCL., Comment: Submitted to Supercond. Sci. Technol

Upper critical fields of several magnesium diboride thin films were measured up to 28 T at the Grenoble High Magnetic Field Laboratory (GHMFL) in Grenoble and up to 60 T at the Laboratoire National des Champs Magnetiques Pulses (LNCMP) in Toulouse. The samples were prepared both by pulsed laser deposition (PLD) and hybrid physical chemical vapour deposition (HPCVD) technique; they have critical temperatures between 29 and 39 K and normal state resistivities between 5 and 250 μohmcm; one of them has been intentionally doped with carbon. The measured critical fields were exceptionally high; we obtained the record value of 52 T at 4.2 K in the parallel orientation. In contrast with the BCS predictions, no saturation in Hc2 at low temperature was observed. Furthermore, films with a wide range of resistivity values showed similar critical fields, suggesting that in a two band system resistivity and Hc2 are not trivially linked. The high Hc2 values seem to be related with the expanded c-axis. The structure of one of the samples was carefully investigated with X-ray diffraction at European Synchrotron Radiation Facility (ESRF) in Grenoble., Comment: 4 pages, 3 figures. Presented at ASC 2004

The Powder-In-Tube process has been employed to fabricate tape-like conductors with a strong metallic sheath and based on the newly discovered MgB2 superconducting phase. Long superconducting tapes have been prepared by packing reacted MgB2 powders inside pure Ag, Cu and Ni tubes which are then cold worked by drawing and rolling. Such tapes have shown transport properties as good as bulk MgB2 samples sintered in high pressure and high temperature conditions. At 4.2 K, the highest critical current density of 100000 A/cm2 has been achieved on Nickel-sheathed single-filament conductors. A direct correlation between the sheath tensile strength and the critical current of the unsintered tape has been observed., Comment: 9 pages, 3 figures; submitted to Appl. Phys. Lett. Received April 3, 2001