Refined fine structure constant

Refined fine structure constant

Physicists from Harvard University, led by Professor Gerald Gabrielse (Gerald Gabrielse), carried out an extremely precise experiment, which made it possible to significantly refine the numerical value
constant fine structure… They published their results in two articles that appeared simultaneously in the journal Physical Review Letters (97, 030801 and 97, 030802). In the first one, the measurement data are presented, in the second, the final calculations.

The fine structure constant – denoted by the Greek letter alpha (α) – was introduced by the German theoretical physicist Arnold Sommerfeld in 1916, before the creation of quantum mechanics. It appeared in Sommerfeld’s calculations describing the doublet splitting of energy levels (and, accordingly, spectral lines) of a hydrogen-like atom in Bohr’s model, due to relativistic effects. This splitting is called the fine structure of the spectrum, hence the name of the constant. Later it turned out that it was caused by the interaction between the orbital and spin moments of the electron, which in itself is a relativistic effect.

In 1916, the concept of spin did not yet exist, and Sommerfeld obtained his results by calculating the energy of an electron up to the square of the ratio of its linear velocity v (which was then still defined purely classically) to the speed of light c, (v/c)2… In these calculations, the fine structure constant was included as the ratio of the speed of an electron in the lower circular orbit to the speed of light. In the CGSE system of units, it is written using a simple formula:

Here e – electron charge, c – the speed of light,

– reduced Planck’s constant, or Dirac’s constant (
= h/ 2π, where h –
Planck’s constant, which relates the magnitude of the energy of electromagnetic radiation with its frequency). α is a dimensionless quantity, its numerical value is very close to 1/137.

The physical meaning of the fine structure constant changed radically after the creation of quantum electrodynamics. In this theory, electrically charged particles interact through the exchange of virtual photons. The fine structure constant appears there as a dimensionless parameter characterizing the intensity of this interaction.

The role of “alpha” is most clearly manifested when calculating various effects using Feynman diagrams, which serve as the main method of approximate calculations in quantum electrodynamics. Each vertex of the Feynman diagram brings a factor equal to the square root of alpha to the numerical value of the amplitude of the calculated process. Since the internal lines that appear in the calculations have two ends, adding each such line gives a factor proportional to alpha. It is due to the smallness of the fine structure constant in quantum electrodynamics that approximate calculations can be made by expanding the calculated values ​​into series in its powers. True, counting some diagrams gives infinities, but in quantum electrodynamics one can get rid of them using the so-called renormalization (however, these are already details).

In the late 1960s, quantum electrodynamics was generalized in the form of a unified theory of electroweak interactions. In this theory, “alpha” grows in proportion to the logarithm of the characteristic energy of the physical process and therefore is no longer a constant. Sommerfeld’s formula corresponds to the limiting value of “alpha” at the lowest possible energies of electromagnetic interaction. Since the lightest electrically charged particles are electrons and positrons, this minimum is reached at an energy equal to the mass of an electron times the square of the speed of light. According to some hypotheses, alpha may also depend on time, but this has not yet been proven.

Quantum electrodynamics does not allow a purely theoretical finding of the specific value of the “force” of electromagnetic interaction. However, it can be established by calculating some physically observable quantity depending on α, and then comparing this result with experiment. This is exactly what Gabriels et al. Did. They used calculations of the internal (spin) magnetic moment of an electron in the fourth order of perturbation theory, which were published this year by Cornell University professor Toichiro Kinoshita and his Japanese colleague Makiko Nio (Physical Review D, 73, 013003, 2006). To calculate the corrections to the value of the magnetic moment published in 1996 in the third order of the perturbation theory, Kinoshite and Nio had to take into account the contributions from the 891 Feynman diagram, which required many years of analytical calculations and calculations on a supercomputer.

As you know, the magnetic moment of an electron is proportional to the product of its spin and Bohr’s magneton. The proportionality coefficient is usually denoted by a Latin letter g… According to the relativistic theory of the electron, formulated in 1928 by Paul Dirac, g = 2. This value was taken on faith for two decades, but in 1948 Polycarp Kush and Henry Foley experimentally proved that g approximately equal to 2.002. At the same time, one of the creators of quantum electrodynamics, Julius Schwinger, obtained the same value theoretically. Quantum electrodynamics explains the excess g-factor over the Dirac value in that the magnetic moment increases due to the creation of virtual particles and the polarization of the vacuum. Since g-factor has been measured more than once experimentally and calculated on the basis of the equations of quantum electrodynamics, and each time the results coincide with ever higher accuracy. In 1987, Hans Demelt and his colleagues measured
g-factor with an accuracy of four trillion, for which two years later Hans Demelt was awarded the Nobel Prize.

The calculations of Kinoshita and Nio made it possible to present g-factor in the form of a finite Taylor series terminating at a term proportional to the fourth power of the fine structure constant α. For experimental verification of this value, the accuracy of the results of the Demelt group was insufficient. Gabriels and his team re-measured g-factor using a device they called a one-electron cyclotron.

Diagram of a one-electron cyclotron used in the experiment.  Image from article PRL 97, 030801 (2006)

Diagram of a one-electron cyclotron used in the experiment. Image from article PRL 97, 030801 (2006)

This device was created by Gabriels and Stephen Payle at the end of the last decade and has been continuously improved since then. It is a small conducting cavity in which a single electron is locked with the help of alternating electromagnetic fields (in fact, this is a modification of a long-known device called the Penning trap). During measurements, a magnetic field is turned on, directed along the axis of the device. The presence of this field makes the electron move in a spiral with the cyclotron frequency fc and simultaneously precess around the field vector with frequency fs

According to the theory, g-factor exceeds two by an amount equal to (fs – fc) / fc … The numerator and denominator of this fraction were determined experimentally. These measurements required an extremely accurate calculation of the geometry of the inner cavity of the trap and its cooling to 0.1 K – all this was necessary to ensure the stability of the electron orbits, since the measurements were carried out over many hours. The experimenters even had to take into account the relativistic corrections, although they were extremely small due to the very low energy of the electron.

Ultimately, the experiment gave meaning
g/ 2 = 1.00115965218085, and the possible error does not exceed 0.76 trillion (that is, the accuracy of the Demelt group is improved sixfold). This value
g-factor allowed us to calculate the value of alpha, which turned out to be equal to 1 / 137.035999710 with an error of the order of 0.7 billionths (tenfold improvement in comparison with previous results).

Such a noticeable refinement of the calculated value of the fine structure constant makes it possible to identify the boundaries of quantum electrodynamics. It is based on the assumption that the electron and positron are point particles. If, as some hypotheses claim, the electron and positron have an internal structure, it should affect the alpha value. (True, the fine structure constant also includes very small additions due to strong and weak interactions, but physicists from the Gabriels group believe that they can be taken into account.)

Now physicists have to again measure the fine structure constant as accurately as possible in other ways (this is done, for example, using solid-state phenomena such as the Josephson effect and the quantum Hall effect, as well as by scattering photons by rubidium atoms) and compare the results obtained with the estimate of the Gabriels group … Who knows what will come of this?


1) B. Odom, D. Hanneke, B. D’Urso, G. Gabrielse. New Measurement of the Electron Magnetic Moment Using a One-Electron Quantum Cyclotron (full text PDF, 256 Kb) // Physical Review Letters, 97, 030801 (2006).
2) G. Gabrielse, D. Hanneke, T. Kinoshita, M. Nio, B. Odom. New Determination of the Fine Structure Constant from the Electron g Value and QED (full text PDF, 200 Kb) // Physical Review Letters, 97, 030802 (2006).
3) Toichiro Kinoshita, Makiko Nio. Improved alpha4 term of the electron anomalous magnetic moment // Phys. Rev. D 73,013003 (2006).

Alexey Levin


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