### Abstract

- Top of page
- Abstract
- 1 Introduction
- 2 Determination of the ratio
- 3 Discussion
- 4 Conclusion
- References

The fine structure constant α has a particular status in physics. Its precise determination is required to test the quantum electrodynamics (QED) theory. The constant α is also a keystone for the determination of other fundamental physical constants, especially the ones involved in the framework of the future International System of units. This paper presents Paris experiment, where the fine structure constant is determined by measuring the recoil velocity of a rubidium atom when it absorbs a photon. The impact of the recent improvement of QED calculations of the electron moment anomaly and the recent measurement of the cesium atom recoil at Berkeley will be discussed. The opportunity to provide a precise value of the ratio between the Planck constant and the atomic mass constant will be investigated.

### 1 Introduction

- Top of page
- Abstract
- 1 Introduction
- 2 Determination of the ratio
- 3 Discussion
- 4 Conclusion
- References

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:

- (1)

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 :

- (2)

- (3)

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 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:

- (5)

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.

### 3 Discussion

- Top of page
- Abstract
- 1 Introduction
- 2 Determination of the ratio
- 3 Discussion
- 4 Conclusion
- References

The experiments which provide the values of the fine structure constant summarized in figure 1 can be used in two different ways. On the one hand, they can be seen as a test of QED calculations of the electron moment anomaly *a*_{e}. These very difficult calculations have been performed by the group of Kinoshita and Nio. They have recently calculated for the first time the fifth coefficient of equation (3) and improved the uncertainty on the fourth one. For the test of QED only two data are involved: the experimental value of *a*_{e}(Exp) (1159652180.73(28)× 10^{−12} [0.28ppt]) achieved by the group of Gabrielse and the one predicted by the theory, *a*_{e}(Theory). The latter is computed using as input data. The disagreement between the experimental and theoretical values of *a*_{e} is:

- (10)

The upper part of figure 7 shows the comparison between the current values of *a*_{e}. The accuracy of the value of is sufficient to test the contributions due to the muon and hadrons in the theoretical value of *a*_{e}. The lower part shows the relative contributions to the electron anomaly of the different terms in Equation (3).

Table 3. The values of the fine structure constant and the ratio deduced from the experiments of Harvard, Berkeley and Paris | [m^{2} s^{−1}] | |
---|

Harvard university | 3.9903127118(26)× 10^{−7} [0.65 ppb] | 137.035999173(35) [0.25 ppb] |

Berkeley university | 3.990312738(16)× 10^{−7} [4.0 ppb] | 137.03599872(28) [2.0 ppb] |

LKB | 3.9903127193(50)× 10^{−7} [1.2 ppb] | 137.035999044(90) [0.66 ppb] |

Reference [13] highlights the impact of such measurements on the proposed redefinition of the SI of units. The interpretation of the aforementioned work needs to be clarified: in the redefinition planned by the CGPM in 2015, the definition of the second will stay the same and the kilogram will be defined by fixing the value of the Planck constant *h*. This definition will be based on fundamental constants and therefore the resolution of the CGPM explicitly relies on the CODATA for the new definition [23].

The main challenge for the redefinition of the Kilogram, and the main reason why this redefinition has been delayed for several years, is the lack of a reliable link between the microscopic and macroscopic masses. This link is established with a relative uncertainty of 3 [28] and with a large discrepancies between the different methods (watt balances and Avogadro project). One can notice that the recently measured value of the Avogadro constant [29], which is the most accurate input datum for the kilogram redefinition, is midway between the watt-balance values [40, 41].

As for the CODATA, the recent measurement of Berkeley is strictly equivalent to an measurement. While it contributes to the reliability of the determination of by providing a determination below 10^{−8}, unfortunately it will not contribute that much to the CODATA (and therefore to the redefinition of the SI) because its uncertainty is too large (see Figure 8, in which is also included the value obtained from α(HarvU), assuming the exactness of QED calculations).

The ratio will have an important role after the redefinition of the SI. As mentioned in reference [42] *This would yield a value for the mass of the atom in SI units, i.e. kilograms, without making reference to the prototype kilogram artefact as is now necessary*. Therefore, there is a strong motivation to continue to improve the uncertainty on as much as possible, until competing methods are obviously superior.

This uncertainty will then be comparable to the ones of the comparison between atomic masses and therefore uncertainties of atomic masses in SI will be the same as in the atomic mass unit (AMU).

### 4 Conclusion

- Top of page
- Abstract
- 1 Introduction
- 2 Determination of the ratio
- 3 Discussion
- 4 Conclusion
- References

In this paper we have presented the details of our recent experimental setup. The fine structure constant is determined with a relative uncertainty of 6.6× 10^{−10}. Taking account of the recent improvement of QED calculations, we deduce a theoretical value of the electron moment anomaly. The comparison with the experimental value of *a*_{e} realized by the group of Gabrielse at Havard university provides the most stringent test of QED.

In the future we plan to improve the accuracy on and therefore on α, by increasing the sensitivity of the atom interferometer (velocity sensor) and by reducing the systematic effect due to the Gouy phase and the wave-front curvature. A new project is currently in progress in our group. It consists on a new experimental setup based on evaporatively cooled atoms. We plan to implement on this setup an atom interferometer based on large momentum beam splitters [44].

This experiment is supported in part by IFRAF (Institut Francilien de Recherches sur les Atomes Froids), and by the Agence Nationale pour la Recherche, FISCOM Project-(ANR-06-BLAN-0192).