Landé g-factor

The Landé factor ( according to Alfred Landé ) (also gyromagnetic factor, in short: g - factor) is an atom, a nucleus or an elementary particle, the ratio of the measured magnetic moment to the magnetic moment, which in the present angular momentum of classical physics would be expected theoretically. A negative factor g indicates that the magnetic moment is opposite to the expected direction.

As a classical reference value, the magnetic moment is calculated for a system having the same mass, the same electric charge and the same angular momentum. For pure orbital angular momentum is a consensus, therefore, is (the index is the symbol for the orbital angular momentum ). Different cases explained if the total angular momentum resulting in whole or in part from the spin.

If there is pure spin angular momentum, ie, the g - factor and spin g - factor or anomalous g - factor of the spin and for each particle has a fixed characteristic value. For example, for the electron, for the proton, for the neutron, for their antiparticles have the same values ​​of opposite sign.

When the total angular momentum of the system of the two types of angular momentum is composed in the considered state, the g-factor is a combination of the landing, and after the formula ( see below).

The system is in a magnetic field, precess vectors ( the total angular momentum ) and ( magnetic moment ), the expected values ​​are always parallel to each other, at the Larmor frequency to the direction of the magnetic field, causing an observable splitting of the energy level by which the g factor can be determined (see the Zeeman effect, nuclear magnetic resonance, electron spin resonance). Such measurements have contributed significantly to the discovery of the spin and to elucidate the structure of the electron shell and the atomic nuclei.

• 2.1 electron
• 2.2 Composite particles
• 2.3 Determination history

Theory

Magnetic moment

According to classical physics has a body with mass and electric charge, which describes the angular momentum of a circular orbit, a magnetic moment

This also applies to the orbital angular momentum in quantum mechanics (here, strictly speaking, the operators and are meant ). The factor is called the gyromagnetic ratio. Analogously, for the spin angular momentum ( which does not exist in classical physics ), but now with the anomalous spin g - factor:

The operators for the total angular momentum and the total magnetic moment of a particle are:

These formulas in general and also for neutral particles to be able to use (like the neutron), although their orbital angular momentum generated because no magnetic moment is chosen constant and writes

Where for particles with charge, and for neutral particles.

If the system consists of several particles of the same kind (eg electrons in the atomic shell ), then all orbital angular momentum operators to the total orbital angular momentum and spin operators all add up to the total spin. The operators for the total angular momentum of the system and its overall magnetic moment are the vector sums

Landé formula

Because of the above defined vectors and not parallel. But if in a set by the energy state of the quantum numbers for the amounts of rail, spin and total angular momentum have (and its z- component) certain values ​​, the magnetic moment acts only through its component parallel to of ( Wigner -Eckart theorem, illustrated by " The angular momentum perpendicular components transmit out. "). To distinguish between the quantum numbers the operators are now " with a roof " ( etc. ) is written. The operator is therefore not a multiple of, but effectively takes its place on the parallel component, with which also the resulting g - factor is defined:

The resulting g - factor is therefore the value of the operator

When squared, the first scalar product can be referred to

Expressed in terms of the quantum numbers, the second analog. It follows the generalized Landé formula

For an electron are employed and thus obtains the usual Landé formula

For a whole atomic shell with multiple electrons, the nature of the coupling of the angular momenta must be considered. Simple Landé formula in the case of LS- coupling correctly, because only in the observed state of the entire orbital angular momentum and the total spin angular momentum have well-defined values ​​. For the calculation, only the valence electrons are taken into account, distributed according to Hund's rules to the various levels of the highest occupied shell, since the angular momentum and spin quantum numbers of closed shells coupled to zero.

The simple Landé formula does not contain the exact spin g - factor of the electron, which is larger by 1.1 ‰ due to effects of quantum electrodynamics.

Landé even in 1923 (almost correctly) specified. Only after the discovery of the quantum-mechanical formulas for the angular momentum of 1925 was the correct version.

Anomalous g - factors of the spin

Electron

In the theoretical description of the electron by the Schrödinger equation, there is no initial spin. With the discovery of half-integer spins the electron due to the observations of the anomalous Zeeman effect of the anomalous gyromagnetic factor had to be attributed. In the extension of the Schrödinger equation for the Pauli equation the spin is included, with the gyromagnetic factor, that is freely selectable unexplained. Only the relativistic description of the electron by the Dirac equation for spin- ½ fermions showed theoretically. Contrary to popular opinion, this value can also be established from the non-relativistic Schrödinger equation, if you modified appropriately. [Note 1]

Experiments of the electron spin resonance showed slight deviations later. Sometimes only these additional deviations are called anomalous magnetic moment experiments for their determination are also called (g- 2 ) experiments. The Dirac equation does not consider the possible creation and annihilation of photons and electron-positron pairs. This is performed until the quantum electrodynamics. Which leads to adjustments in the coupling of the electron to the magnetic field. They provide a theoretical value of

Whereas experiments according to the current measurement accuracy a value of

Result. It is noteworthy that the value is experimentally known with higher accuracy than he can be calculated theoretically. The precise calculation of the g factor and the comparison with the experiment about the muon is used for precision tests of the Standard Model of elementary particles.

Composite particles

Together laws particles have significantly different gyromagnetic factors:

The g - factors of the nucleons are not exactly predictable, since the behavior of its constituents, quarks and gluons, is not known with sufficient accuracy.

When gyromagnetic factor of the neutron is taken exactly at the strength of the spin- magnetic-field energy of the neutron compared to the orbital angular momentum magnetic field energy of the proton, because the neutron is uncharged and has no orbital angular momentum magnetic field energy.

As well as the gyromagnetic factors of protons and neutrons, the core -g- factor can not be calculated a priori but must be determined experimentally.

Determination history

The g- factor, in particular, the value for the electron, in 1923 phenomenologically introduced by Landé, to summarize the observations of the anomalous Zeeman effect in formulas. A theoretical explanation was established in 1928 with the Dirac equation have been found. The outlying values ​​for proton (1933 ) and neutron (1948 ) could be understood until decades later in the quark model. The small deviation from the Dirac value of the electron was in bound electrons by Polycarp Kusch and others from 1946 discovered for free electrons by H. Richard Crane from 1954, to a precision measurement at 13 decimal places 2011 by D. Hanneke, in accordance with the theoretical calculation in the standard model of elementary particles

The determination of the g-factor of the muon focused particularly Vernon Hughes, culminating in an experiment at Brookhaven National Laboratory, the results of which were presented in 2002. The comparison with the theory is the muon insofar difficult as incorporated in the theoretical value of additional experimental values ​​with less precision with. A 2009 analysis showed a deviation from the predictions of the Standard Model.