## 1 Introduction

Since its discovery at the beginning of the 20 century up to nowadays, the fine structure constant α remains one of the most fascinating fundamental constants, as it is dimensionless. Currently it plays a central role in the Physics of the 21 century by testing the most accurate theories such as quantum electrodynamics (QED) [1-3], testing the stability of fundamental constants () (for example see review by J.P. Uzan [4]) but also in a practical way in the proposed redefinition of the international system of units (SI) [5].

The name of the fine structure constant derives from the Sommerfeld model [6]. It was intended to explain the fine structure of the hydrogen spectral lines, unaccounted for in the Bohr model. The Sommerfeld model combines the theory of relativity with the Bohr model. The constant α appears in the velocity of the electron (*v*_{e}) on its first orbit around the proton (, where *c* is the velocity of light). The expression for α is:

where *e* is the charge of the electron, ε_{0} the vacuum permittivity and in which *h* is the Planck constant.

The Sommerfeld model failed because it didn't take into account the spin of the electron. Nevertheless the constant introduced in this model is still relevant in the Dirac model which combines relativity and quantum mechanics [7]. This model predicts the existence of the positron and the spin of the electron! In 1947 a new effect from which the value of α can be deduced was discovered: the vacuum quantum fluctuations which contribute to the splitting of and energy levels in hydrogen (now usually called the Lamb shift) [8, 9] and also contribute to the anomaly of the gyromagnetic factor of leptons [10, 11].

Indeed the modern understanding of α is that it sets the scale of the electromagnetic interaction. Consequently many experiments in which a charged particle interacts with an electromagnetic field can be used to determine α. In 1998, the experiments considered by the CODATA task group on fundamental constants to give the best estimate of the fine structure constant value ranged from solid state physics and atomic physics to quantum electrodynamics [12].

As shown in figure 1, the current most precise determination of the fine structure constant comes mainly from two methods.

The first method combines the measurement of the electron magnetic moment anomaly *a*_{e} and QED perturbation theory. The value of α is determined by comparing the experimental value of *a*_{e} with :

Thus in the QED model *a*_{e} is expressed as a power series of α and an additive term which takes into account the contributions due to the muon, the tau, the weak and hadronic interactions. The coefficients are finite and dimensionless constants calculated by using Feynman diagrams [3].

The second one, introduced by the group of S. Chu at Stanford university [15], is based on the measurement of the ratio , between the Planck constant *h* and the atomic mass *m*_{X}. This ratio is related to α by

The Rydberg constant is known with an accuracy of [16-18]. The uncertainty on the relative mass of the electron and the relative atomic mass are respectively [19] and less than 10^{−10} for Rb and Cs [20, 21]. Using an atom interferometer and Bloch oscillations, we have performed in 2010 a determination of the ratio . The value that has been deduced is the most precise value obtained using this method[1].

The comparison of these two determinations is one of the most precise test of QED. It is so accurate that one can think, in a near future, of using these lab-size experiments to check theoretical predictions tested up to now only on particle accelerators (for example the existence of internal structure of the electron [22]).

For many years, the main contribution to the determination of α_{CODATA} has been the one derived from the anomaly of the gyromagnetic factor of the electron (α(a_{e})) which is strongly dependent on complex QED calculations. Nowadays the uncertainties of α(a_{e}) and α(Rb) are in the same order of magnitude. This makes the CODATA adjustment more reliable.

This reliability is essential for the redefinition of the SI which will rely on the values of fundamental constants [23-25]. In the proposed redefinition, the Planck constant will have a fixed value in SI units[5]. In order to link the microscopic definition to the macroscopic Kilogram, two kinds of experiments are competitive. The first one, the watt balance measures the ratio between the Planck constant and a macroscopic standard mass *M* [26-28]. In the current SI, it gives a determination of *h*. In the future SI, it will give the measurement of a macroscopic mass. The second experiment is the Avogadro project, which directly determines the ratio between a macroscopic mass (the mass of a silicon sphere) and a microscopic mass (the mass of the atom of silicon) [29]. In the current SI, it gives a determination of the (unified) atomic mass constant *m*_{u} defined according to *m*_{u} = *m*(^{12}C)/12, or the Avogadro constant. The ratio provides therefore a direct comparison between the two experiments. Its precise determination has a major interest in metrology. Whereas the photon-recoil measurement, combined with the appropriate relative atomic mass measurement, gives a determination of the ratio , other values of α can be converted into using the formula:

We emphasize that the ratio becomes identified with Avogadro Planck constant as:

where M(^{12}C) = 12 kg/mol is the carbon molar mass and *N*_{A} is the Avogadro constant. The product is in the current SI, equivalent to the ratio . It seems to us more relevant to consider in the framework of the redefinition of the kilogram. In the future SI of units, the Avogadro constant *N*_{A}, which is used by the chemists to quantify and identify an amount of substance with atoms and molecules, will be fixed. This will break the link between atomic masses and molar masses. Consequently *M*(^{12}C) will no longer be equal to 12 g/mol, but will be determined from equation (6) using the ratio .

In the proposed new International Systems of Units, many others physical constants, that are set by the CODATA will have a fixed value. The constant α will be a keystone of the proposed SI, as many of the remaining constants will depend strongly on its knowledge (such as the vacuum permeability μ_{0}, the von Klitzing constant *R*_{K}, ...)[5].

The next and largest section of this paper will be devoted to the experiment in Paris. This experiment started in 1998 and was entirely renewed in 2008. In the last part, we will discuss the role of the various determinations of α. We will focus on the test of QED calculations and on the impact on the redefinition of the Kilogram.