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# Physikalisch-Technische Bundesanstalt

StructureDiv. 2 Electricity2.6 Electrical Quantum Metrology2.62 Quantum Hall Effect, Resistance > Ohm realization, QHE, Fine-structure constant
Ohm realization, basic research on the quantum Hall effect, and determination of the fine-structure constant

The unit of the electrical resistance is the ohm. The ohm is defined as that resistance which produces a voltage drop of 1 V at a current of 1 A. The realisation of the resistance unit on the basis of the SI units is rather sophisticated. The smallest uncertainty is achieved nowadays if the realization of the resistance unit takes place via the capacitance unit Farad. This is realized by means of a calculable capacitor on which a length measurement has to be performed. This realization of the resistance unit can be done by only a few larger national metrological institutes besides PTB.

Since 1990 resistance standards are calibrated world-wide standardised on the basis of the quantum Hall effect (QHE). By means of the QHE quantised resistance values Ri can be reproduced, the values of Ri are given by Ri = (1/i) RK. RK is the von-Klitzing constant. for which the relation RK = h/e2 is valid (e charge of the electron, h Planck´s constant, i is an integer number). This theoretical statement is in accordance with all up to now reported experimental results for RK. The value of RK is about 25812 ohm. The quantised resistance values Ri can be reproduced experimentally with a relative uncertainty of about 1 part in 109.

In physics the fine-structure constant α is of great importance. The value of this dimensionless quantity is about 1/137 and it was introduced by Sommerfeld as:

α -1 = (2/µ0c)(h/e2)

(µ0 magnetic field constant, c velocity of light in vacuum). On condition that RK = h/e2 each determination of the von-Klitzing constant is a determination of the fine-structure constant, too, and vice versa:

α -1 = (2/µ0c)RK

Besides those values for α which have been got from QHE experiments there are further values which have been deduced from neutron experiments and from quantum electrodynamic theory. All these values are in agreement within a relative uncertainty of 50 parts in 109. This is a very impressive demonstration that the physical models which are used to describe these different experiments are consistent with each other. But it means, too, that the basic equation of the quantum Hall effect is correct within the mentioned uncertainty:

RK = h/e2

© Physikalisch-Technische Bundesanstalt, last update: 2015-01-27, Susanne Gruber