Novel Boron-10-based detectors for Neutron Scattering Science

Nowadays neutron scattering science is increasing its instrumental power. Most of the neutron sources in the world are pushing the development of their technologies to be more performing. The neutron scattering development is also pushed by the European Spallation Source (ESS) in Sweden, a neutron facility which has just started construction. Concerning small area detectors (1m^2), the 3He technology, which is today cutting edge, is reaching fundamental limits in its development. Counting rate capability, spatial resolution and cost-effectiveness, are only a few examples of the features that must be improved to fulfill the new requirements. On the other hand, 3He technology could still satisfy the detector requirements for large area applications (50m^2), however, because of the present 3He shortage that the world is experiencing, this is not practical anymore. The recent detector advances (the Multi-Grid and the Multi-Blade prototypes) developed in the framework of the collaboration between the Institut Laue-Langevin (ILL) and ESS are presented in this manuscript. In particular two novel 10B-based detectors are described; one for large area applications (the Multi-Grid prototype) and one for application in neutron refectometry (small area applications, the Multi-Blade prototype)

An optimal bound for nonlinear eigenvalues and torsional rigidity on domains with holes

In this paper we prove an optimal upper bound for the first eigenvalue of a Robin-Neumann boundary value problem for the p-Laplacian operator in domains with convex holes. An analogous estimate is obtained for the corresponding torsional rigidity problem

A saturation phenomenon for a nonlinear nonlocal eigenvalue problem

Given $1\le q \le 2$ and $\alpha\in\mathbb R$, we study the properties of the solutions of the minimum problem $$\lambda(\alpha,q)=\min\left\{\dfrac{\displaystyle\int_{-1}^{1}|u'|^{2}dx+\alpha\left|\int_{-1}^{1}|u|^{q-1}u\, dx\right|^{\frac2q}}{\displaystyle\int_{-1}^{1}|u|^{2}dx}, u\in H_{0}^{1}(-1,1),\,u\not\equiv 0\right\}.$$ In particular, depending on $\alpha$ and $q$, we show that the minimizers have constant sign up to a critical value of $\alpha=\alpha_{q}$, and when $\alpha>\alpha_{q}$ the minimizers are odd

On the second Dirichlet eigenvalue of some nonlinear anisotropic elliptic operators

Let $\Omega$ be a bounded open set of $\mathbb R^{n}$, $n\ge 2$. In this paper we mainly study some properties of the second Dirichlet eigenvalue $\lambda_{2}(p,\Omega)$ of the anisotropic $p$-Laplacian $$-\mathcal Q_{p}u:=-\textrm{div} \left(F^{p-1}(\nabla u)F_\xi (\nabla u)\right),$$ where $F$ is a suitable smooth norm of $\mathbb R^{n}$ and $p\in]1,+\infty[$. We provide a lower bound of $\lambda_{2}(p,\Omega)$ among bounded open sets of given measure, showing the validity of a Hong-Krahn-Szego type inequality. Furthermore, we investigate the limit problem as $p\to+\infty$

A sharp weighted anisotropic Poincar\'e inequality for convex domains

We prove an optimal lower bound for the best constant in a class of weighted anisotropic Poincar\'e inequalitie

The Multi-Blade: The 10B-based neutron detector for reflectometry at ESS

Abstract The Multi-Blade detector has been designed to be used on the reflectometry instruments at the upcoming European Spallation Source. It is a 10B-based gaseous detector, built as a modular stack of multi-wire proportional chambers organised on a circle around the sample. The detector has been fully characterised. The gamma and fast-neutron sensitivity has been measured at the Source Testing Facility in Lund University, Sweden; the working capability in a reflectometry instrument has been demonstrated with measurements at CRISP in ISIS, UK; and the count-rate capability of the detector will be measured the summer of 2018 at the Budapest Neutron Centre, Hungary

Sharp estimates for the first Robin eigenvalue of nonlinear elliptic operators

The aim of this paper is to obtain optimal estimates for the first Robin eigenvalue of the anisotropic $p$-Laplace operator, namely: \begin{equation*} \lambda_1(\beta,\Omega)=\min_{\psi\in W^{1,p}(\Omega)\setminus\{0\} } \frac{\displaystyle\int_\Omega F(\nabla \psi)^p dx +\beta \displaystyle\int_{\partial\Omega}|\psi|^pF(\nu_{\Omega}) d\mathcal H^{N-1} }{\displaystyle\int_\Omega|\psi|^p dx}, \end{equation*} where $p\in]1,+\infty[$, $\Omega$ is a bounded, mean convex domain in $\mathcal R^{N}$, $\nu_{\Omega}$ is its Euclidean outward normal, $\beta$ is a real number, and $F$ is a sufficiently smooth norm on $\mathcal R^{N}$. The estimates we found are in terms of the first eigenvalue of a one-dimensional nonlinear problem, which depends on $\beta$ and on geometrical quantities associated to $\Omega$. More precisely, we prove a lower bound of $\lambda_{1}$ in the case $\beta>0$, and a upper bound in the case $\beta<0$. As a consequence, we prove, for $\beta>0$, a lower bound for $\lambda_{1}(\beta,\Omega)$ in terms of the anisotropic inradius of $\Omega$ and, for $\beta<0$, an upper bound of $\lambda_{1}(\beta,\Omega)$ in terms of $\beta$.Comment: 24 page