
Introduction to the constants for nonexperts
Current advances: The finestructure constant and quantum Hall effect The finestructure constant α is of dimension 1 (i.e., it is simply a number) and very nearly equal to 1/137. It is the "coupling constant" or measure of the strength of the electromagnetic force that governs how electrically charged elementary particles (e.g., electron, muon) and light (photons) interact. Currently, the value of α having the smallest uncertainty comes from the comparison of the theoretical expression a_{e}(theor) and experimental value a_{e}(expt) of the anomalous magnetic moment of the electron a_{e}. Starting in the 1980's, a new and wholly different measurement approach using the quantum Hall effect (QHE) has caused excitement because the value of α obtained from it independently corroborates the value of α from the electron magnetic moment anomaly. The QHE value of α does not have as small an uncertainty as the electron magnetic moment value, but it does provide a significant independent confirmation of that value. The quantity α was introduced into physics by A. Sommerfeld in 1916 and in the past has often been referred to as the Sommerfeld finestructure constant. In order to explain the observed splitting or fine structure of the energy levels of the hydrogen atom, Sommerfeld extended the Bohr theory to include elliptical orbits and the relativistic dependence of mass on velocity. The quantity α, which is equal to the ratio v_{1}/c where v_{1} is the velocity of the electron in the first circular Bohr orbit and c is the speed of light in vacuum, appeared naturally in Sommerfeld's analysis and determined the size of the splitting or finestructure of the hydrogenic spectral lines. Sommerfeld's theory had some early success in explaining experimental observations but could not accommodate the discovery of electron spin. Although the Dirac relativistic theory of the electron introduced in 1928 solves the main aspects of the problem of the hydrogen finestructure, α still determines its size as in the Sommerfeld theory. Consequently, the name "finestructure" constant for the group of constants below has remained: , 
where e is the elementary charge, = h/2π where h is the Planck constant, ε_{0} = 1/µ_{0}c^{2} is the electric constant (permitivity of vacuum) and µ_{0} is the magnetic constant (permeability of vacuum). In the International System of Units (SI), c, ε_{0}, and µ_{0} are exactly known constants.
Our view of the finestructure constant has changed markedly since Sommerfeld introduced it over 80 years ago. We now consider α the coupling constant for the electromagnetic force and similar to those for the other three known fundamental forces or interactions of nature: the gravitational force, the weak nuclear force, and the strong nuclear force. Further, since α is proportional to e^{2}, it is viewed as the square of an effective charge "screened by vacuum polarization and seen from an infinite distance." According to quantum electrodynamics (QED), the relativistic quantum field theory of the interaction of charged particles and photons, an electron can emit virtual photons that can then emit virtual electronpositron pairs (e^{+}, e^{}). The virtual positrons are attracted to the original or "bare"
electron while the virtual electrons are repelled from it. The bare electron is therefore screened due to this polarization. The usual finestructure constant α is defined as the square of the completely screened charge, that is, the value observed at infinite distance or in the limit of zero momentum transfer. At shorter
distances corresponding to higher energy processes or probes (large momentum transfers), the screen is partially penetrated and the strength of the electromagnetic interaction increases since the effective charge increases. Thus α depends upon the energy at which it is measured, increasing with increasing energy, and is
considered an effective or running coupling constant. Indeed, due to e^{+} e^{} and other vacuum polarization processes, at an energy corresponding to the mass of the W boson As indicated above, the value of alpha from the quantum Hall effect (QHE) has corroborated the value from the electron magnetic moment anomaly a_{e}. The QHE is characteristic of a completely quantized twodimensional electron gas. Such a gas may be realized in a highmobility semiconductor device such as a silicon metaloxidesemiconductor fieldeffect transistor (MOSFET) or GaAsAl_{x}Ga_{1x }As heterojunction of standard Hall bar geometry in an applied magnetic flux density B of the order of 10 T and cooled to about 1 K. For a fixed current I (typically 10 µA to 50 µA) through the device, there are regions in the curve of Hall voltage U_{H }versus gate voltage for a MOSFET, or of U_{H }vs B for a heterojunction, where U_{H }remains constant as either the gate voltage or B is varied. These regions of constant U_{H }are termed quantum Hall plateaus. In the limit of zero dissipation (zero voltage drop) in the direction of current flow, the Hall voltagetocurrent quotient U_{H}(i)/I or Hall resistance R_{H}(i) of the ith plateau, where i is an integer (we consider only the integral QHE), is quantized and given by The theory of the QHE predicts, and the experimentally observed universality of R_{H}(i) = U_{H}(i)/I = R_{K}/i is consistent with the prediction, that R_{K }= h/e^{2}= µ_{0}c/2α. Since in the SI µ_{0} = 4π In practice, R_{K }is measured in terms of a laboratory standard of resistance. Thus, the resistance of the standard must be determined in the SI unit ohm in a separate experiment using an apparatus known as a calculable cross capacitor in which the unknown resistance of a reference resistor is compared with the known impedance of the capacitor. The change in capacitance of such a capacitor, and hence its change in impedance, can be readily calculated since the change depends only on the position of a movable screen electrode whose displacement can be measured with a laser interferometer. In the NIST version of the experiment, the known 0.5 pF change in capacitance of the NIST calculable cross capacitor is used to measure the capacitances of 10 pF reference capacitors. These and a 10:1 bridge are then used in two stages to measure the capacitance of two 1000 pF capacitors, which are in turn used as two arms of a special frequency dependent bridge to measure the impedances of two 100 kiloohm resistors. The latter are then compared using a 100:1 bridge with a 1000 ohm transportable resistor, which in turn is compared using dc techniques with the resistance standard in terms of which R_{K }has been measured. The acdc resistance difference of the 1000 ohm resistor is determined by means of a special 1000 ohm coaxial resistor of negligible acdc resistance difference. All ac measurements are carried out at a frequency of approximately 1592 Hz (2πf = 10^{4} rad/s). The QHE has already yielded a value of α with a relative standard uncertainty of



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