Recalls on the magnetic Moment of spin

Definition. Factor of Moor

At the orbital kinetic time of a particle of load $q$ and mass $m$ is associated a magnetic Moment orbital:

The factor $q/2m$ is called gyromagnetic Rapport. In the same way, one associates with a particle of load $q$, mass $m$, and Spin given a magnetic moment of spin :

where $g$ is a pure number, called Facteur of Moor (1921). This number varies according to the nature of the particle: there is roughly $g=2$ for the electron, $g=2.793$ for the proton, and $g=-\; \backslash ,\; 1.913$ for the neutron.

Magneton of Bohr

For the electron, one has the values following: $s=\; \backslash \; hbar\; /2$ and $g=2$; one then introduces the “magnetic quantum” following, called magneton of Bohr :

Abnormal magnetic moment of the electron

The equation of Dirac predicted for the electron a factor of Moor exactly equal to: $g=2$. However, the allowed experimental value in 2005 is worth:

There thus exists a variation, detected for the first time in 1947 in the hyperfine structure of hydrogen and deuterium.

Anomaly

One is thus brought to introduce a anomaly $a$, defined by:

The Quantum theory of the fields of the standard Modèle makes it possible to calculate this anomaly. The dominant contribution comes from the quantum electrodynamic pertubative, and appears as a development in series of powers of the Constante of fine structure $\backslash \; alpha$, also called constant of coupling . More precisely, one is brought has to write the following development:

in powers of $\backslash \; alpha\_1\; =\; \backslash \; alpha/\backslash \; pi\; \backslash \; simeq\; \backslash \; 0.002\; \backslash \; 322\; \backslash \; 819\; \backslash \; 465\; \backslash \; 36$.

First correction of Schwinger

The first term of the development, calculated by Schwinger in 1948, is worth simply: $A\_1\; =\; 1/2$. It was the first great success of the any quantum news electrodynamic. This calculation, which rests on the Diagramme of Feynman opposite, is today a standard exercise for any postgraduate student beginning in Quantum theory from the fields.

Unfortunately, calculations of the following terms are complicated much, because the number of diagrams believes exponentially quickly with the order of the development.

Correction of order two

This calculation utilizes 7 diagrams of Feynman. A first result - erroneous - was published in 1950, then re-examined and corrected in 1957-1958. One obtains:

where $\backslash \; zeta\; (S)$ is the Fonction zeta of Riemann, defined by:

and checking in particular: $\backslash \; zeta\; (2)\; =\; \backslash \; pi^2/6$.

Correction of order three

This calculation utilizes 72 diagrams of Feynman. The calculation, started in 1969, was finished and published only in 1996. A complicated analytical expression is obtained, which one will find for example in p 101. Numerically, one obtains:

Correction of order four

This calculation, which utilizes 891 diagrams of Feynman, is impossible to make entirely with the hand in a reasonable time! It required the intensive use of the computer. The best numerical result, published in 1999, is:

Comparison theory - experiment

The electron being the lightest lepton, the contributions to its magnetic moment of the other leptons, the bosons vectors of the weak interaction, and the quarks and let us gluons, are small, but considerable with the current precision. Their inclusions gives the theoretical prediction of the standard Modèle:

The agreement with the results experimental is to date excellent:

See too

• magnetic Spin

• Moment of quantum spin
• kinetic Moment
• orbital kinetic Moment
• Quantum theory of the fields