Table of contents

The quantum Hall effect (QHE) provides an invariant reference for resistance linked to natural constants. It is used worldwide to maintain and compare the unit of resistance. The reproducibility reached today is almost two orders of magnitude better than the uncertainty of the determination of the ohm in the international system of units SI.

In this article, mainly the aspects of the QHE relevant for its metrological application are reviewed. After a short introduction of the theoretical models describing the integer QHE, the properties of the devices used in metrology and the measurement techniques are described. A detailed summary is given on the measurements carried out to demonstrate the universality of the quantized Hall resistance and to assess all the effects leading to deviations of the Hall resistance from the quantized value. In addition, the present and future role of the QHE in the SI and the field of natural constants is discussed.

Nuclear Astrophysics is concerned with the study of nuclear processes at stellar temperature and density conditions. A main goal is the understanding of the synthesis of the elements and the generation of energy guiding stellar evolution and driving stellar explosions. Observables (like e.g. luminosity curves or elemental abundance distributions) witness the interplay between nuclear structure aspects near the particle drip-lines and the appropriate astrophysical environments, and give guidance to and constraints on stellar conditions and the associated nucleosynthesis. We present an overview of the broad range of nucleosynthesis scenarios from the Big Bang to the different conditions during stellar evolution and stellar explosion. Special emphasis is given to the discussion of nuclear physics aspects of pre-supernova collapse and supernova shock front nucleosynthesis. Of great interests are presently nucleosynthesis processes far from the limits of stability like the neutron diriven r-process and the hydrogen driven rp-process. The nuclear physics of the r-process and the possible site for the r-process in the neutrino driven wind of the supernova shock are discussed and the possible impact of neutrino induced processes is presented. Hydrogen induced explosive processes occur in the thermonuclear runaway on the surface of accreting compact stars at electron degenerate conditions. This includes novae triggered by accretion on white dwarfs and x-ray bursts initiated by accretion on neutron stars. High accretion rates on white dwarfs and neutron stars lead to supernova type Ia explosions or to x-ray pulsars respectively. An overview of the nucleosynthesis conditions and uncertainties is given for all of these scenarios.

This is an introduction to spin foam models for non-perturbative quantum gravity, an approach that lies at the point of convergence of many different research areas, including loop quantum gravity, topological quantum field theories, path integral quantum gravity, lattice field theory, matrix models, category theory and statistical mechanics. We describe the general formalism and ideas of spin foam models, the picture of quantum geometry emerging from them, and give a review of the results obtained so far, in both the Euclidean and Lorentzian cases. We focus in particular on the Barrett-Crane model for four-dimensional quantum gravity.

The basics of magnetoencephalography (MEG), i.e. the measurement and the analysis of the tiny magnetic fields generated outside the scalp by the working human brain, are reviewed. Three main topics are discussed: (1) the relationship between the magnetic field and its generators, including on one hand the neurophysiological basis and the physical theory of magnetic field generation, and on the other hand the techniques for the estimation of the sources from the magnetic field measurements; (2) the instrumental techniques and the laboratory practice of neuromagnetic field measurement and (3) the main applications of MEG in basic neurophysiology as well as in clinical neurology.

One of the most significant developments in computational atomic and molecular physics in recent years has been the introduction of B-spline basis sets in calculations of atomic and molecular structure and dynamics. B-splines were introduced in applied mathematics more than 50 years ago, but it has been in the 1990s, with the advent of powerful computers, that the number of applications has grown exponentially. In this review we present the main properties of B-splines and discuss why they are useful to solve different problems in atomic and molecular physics. We provide an extensive reference list of theoretical works that have made use of B-spline basis sets up to 2000. Among these, we have focused on those applications that have led to the discovery of new interesting phenomena and pointed out the reasons behind the success of the approach.

The Avogadro constant, N_A, is a fundamental physical constant that relates any quantity at the atomic scale to its corresponding macroscopic scale. Inspired by the kinetic gas theory Avogadro proposed his hypothesis in 1811, in order to describe chemical reactions as an atomic process between atoms or molecules. Starting from his pioneering findings, the determination of this large number has fascinated generations of scientists up to this day. The review of methods aimed at finding a value for N_A starts with the calculations made by Loschmidt (1865; N_A ≈72×10²³ mol^-1) who evaluated the number of molecules in a given gas volume, derived from estimates of molecular diameters and the mean free path length. Consideration of Brownian motion led to some more accurate determinations of N_A around the beginning of the 20th century (Perrin (1908); N_A≈6.7×10²³ mol^-1). Other methods developed in the following years are based on Millikan's oil drop experiment (1917, N_A≈6.064(6)×10²³ mol^-1), on the counting of alpha particles emitted from radium or uranium (Rutherford (1909); N_A≈6.16×10²³ mol^-1) and on investigations of molecular monolayers on liquids (Nuoy (1924); N_A≈6.004×10²³ mol^-1).

A modern method to derive N_A from the density, the relative atomic mass, and the unit cell length was introduced by Bragg in 1913. It makes use of the diffraction of x-rays by the interatomic spacings of a crystal lattice and its periodic arrangement. The accuracy of this method is extremely affected by the fact that the lattice scale of the structurally imperfect lattice can be calibrated only approximately in SI units. Data of N_A were, therefore, found to be in disagreement with other fundamental constants (Bearden (1931); N_A≈6.019(3)×10²³ mol^-1). A break though was achieved with perfect crystals of silicon and x-ray interferometry making available very precise data of atomic distances, expressed in SI units (Bonse and Hart 1965).

Today, metrology has re-discovered the Avogadro constant and uses it as one of several possible routes to a re-definition of the kilogram because the old platinum iridium artefact exhibits long-term stability problems. This application of the Avogadro constant presupposes a final measurement uncertainty of about 1×10^-8, a challenge for the experimental determination of the quantities involved, i.e. macroscopic density, isotopic composition, and unit cell volume of a silicon crystal. Many years of research work were centred on the problem of how far the perfection of a real crystal is away from the ideal state. At present, it is widely accepted that, in the limits of the desired uncertainty, the lattice parameter, and thus the unit cell volume of silicon, can be seen as an invariant quantity when the influence of residual defects, for example impurities, is taken into account. Up to a relative measurement uncertainty of a few parts in 10⁷ it has recently been shown that the molar volume, the ratio of molar mass to density, is constant, too. The combination of data from several independent measurements of the unit cell and the molar volumes has led to a value for the Avogadro constant of N_A = 6.022 1335(30)×10²³ mol^-1 (De Bièvre et al 2001) recommended by the national metrology institutes involved in this research project (Becker 2001).

Prominent examples of the significance of the research work reviewed here are the use of N_A as an input independent of other data, for the adjustment of a consistent set of fundamental constants, and the accompanying outstanding experimental developments acting as spin-offs in the field of technology to make macroscopic dimensions traceable to the atomic scale.