Determination of the fine-structure constant a
by measuring the quotient
of the Planck constant and the neutron mass

E. Krüger, W. Nistler, W. Weirauch (PTB Braunschweig).

Using a special high-precision apparatus at ILL the quotient h/mn (h Planck constant, mn neutron mass) has been measured with a relative uncertainty of 7.7.10-8. The value measured for h/mn leads to a-1 = 137.03601082(524) (relative uncertainty: 3.9.10-8 ). It was the first time that this fundamental constant has been determined by means of neutrons. The experiment, which had been running since 1981 in a preliminary version and since 1987 in the final version, was finished in December 1996.

On the basis of the atomic model created by Bohr it had been possible for the first time to understand the energy levels of the hydrogen atom. Their fine structure was explained by Sommerfeld's theory which expanded Bohr's model.

After the introduction of quantum mechanics, Sommerfeld's theory was replaced by the theory of Dirac which also took into account the spin of the hydrogen atom. The fine-structure constant a here is of central importance.

Nowadays it is interpreted as the coupling constant of the electromagnetic interaction. Its exact value is of great importance for the system of fundamental constants. Several methods are used to measure a with high precision:

1. Measurement of the magnetic moment of the electron. This method needs a high degree of quantum electrodynamical calculations, but is the most precise method today.

2. Measurement of the von Klitzing constant RK, i.e. the height of the resistance steps in the quantum Hall effect. The theory of this effect furnishes a relation between a and RK.

To achieve high accuracy, the SI unit "ohm" must be determined by means of a calculable capacitor.

3. Measurement of the gyromagnetic ratio of the proton. The Larmor precession of the proton spin inside the magnetic field of a precise coil is measured. Additional constants needed for the evaluation of a from the measured data are very precisely known.

4. Measurement of the muonium hyperfine structure. The splitting of the ground level leads to a value of a by means of quantum electrodynamical calculations.

5. Measurement of the quotient of Planck constant and neutron mass. The velocity and the wavelength of thermal neutrons are determined and according to de Broglie this gives h/mn. In order to measure the velocity, the polarisation vector of polarised neutrons is modulated in a special time-of-flight method by a high-frequency magnetic field. The wavelength of the neutrons is determined by back-reflection in a nearly perfect silicon crystal. Details of the measuring arrangement can be found.

Figure 1: Apparatus for measuring h/mn, installed in the neutron guide hall of the ILL.

The relation between a and h/mn is as follows: a = [(2R¥/c)(Mn/Me)(h/mn)]1/2. (R¥ Rydberg constant, c speed of light in vacuum, Mn, Me relative atomic masses of neutron and electron.) The quantities needed in addition to h/mn are very precisely known.

The method for measuring h/mn was proposed by one of the authors in 1977. Preliminary investigations were carried out at the reactor of the Physikalisch-Technische Bundesanstalt (PTB) in Braunschweig. They were followed by detailed test measurements at the ILL in 1981. Finally, the high-precision experiment was installed in 1987 (see Fig. 1). Measurements were done until the reactor shutdown in 1991. After its restart in 1995 they were continued until the end of 1996.

Figure 2 (left): Values of h/mn. Empty circles: measured values, in chronological order from top to bottom; solid circle: weighted mean of the values measured before April 1991. The latter, denoted in this figure by (h/mn )91 is used as a reference value.
Figure 3 (right): Values of the reciprocal fine-structure constant a-1. Methods: CODATA 1986: least-squares adjustment of the fundamental constants in 1986; Mhfs: measurement of the muonium hyperfine structure; g'p: measurement of the gyromagnetic ratio of the proton; RK: measurement of the von-Klitzing constant by means of the quantum Hall effect; ae: measurement of the magnetic moment of the electron; h/mn: this experiment. The reference value a&endash;190 is calculated from the defined value RK-90 of the von-Klitzing constant.

Many measurements were carried out with various alter-ations of the measuring parameters. Fig. 2 shows the results. Four crystals from two silicon bars were used in the measurements. They have different lattice parameters as the content of carbon impurities differs. So far, only the lattice parameter of Si1 has been measured at the PTB by means of X-rays. It is intended to measure the other crystals during the final evaluation of the h/mn measurement. In Fig. 2 it is assumed for the moment that Si4 and Si5 are perfect crystals and the lattice parameter of Si2 equals that of Si1.

The evaluation of the values measured before 1991 resulted in h/mn = 3.956 033 320 (302) . 10-7 m2s-1 with a relative uncertainty of 7.7 . 10 -8. It follows that a-1 = 137.036 010 82 (5 24) the relative uncertainty being 3.9 . 10 -8. The present situation for a-1 is shown in Fig. 3. The differences between the measured values are greater than expected in view of their uncertainties. The discrepancies could not be clarified so far. Nevertheless, the value of CODATA tabulated in 1986 seems to be too low.