# Landé factor

The Landé factor (after Alfred Landé ) (also gyromagnetic factor , short: g-factor ) is for an atom , an atomic nucleus or an elementary particle the quotient of the magnitude of the measured magnetic moment and the magnitude of the magnetic moment that is present in the present Angular momentum would theoretically be expected according to classical physics . The sign indicates whether the magnetic moment is parallel or antiparallel to the expected direction. Here, however, the convention is not entirely clear, so that the same g-factor can be found in the literature with different signs. ${\ displaystyle g}$

As a classic comparison value, the magnetic moment is calculated for a system that has the same mass , the same electrical charge and the same angular momentum . With pure orbital angular momentum there is agreement, therefore (the index is the symbol for the orbital angular momentum). Deviating cases can be explained if the total angular momentum originates entirely or partially from the spin . ${\ displaystyle g _ {\ ell} = 1}$${\ displaystyle \ ell}$${\ displaystyle g {\ neq} 1}$

If there is pure spin angular momentum, the g-factor is also called the spin g-factor or anomalous g-factor of the spin and has a fixed characteristic value for each type of particle. For example is for the electron , for the proton , for the neutron . ${\ displaystyle g_ {s}}$ ${\ displaystyle g_ {s} \ approx 2}$ ${\ displaystyle g_ {s} \ approx 5 {,} 6}$ ${\ displaystyle g_ {s} \ approx -3 {,} 8}$

If the total angular momentum of the system in the considered state is composed of both types of angular momentum, the g-factor is a combination of and according to the Landé formula (see below). ${\ displaystyle g_ {s}}$${\ displaystyle g _ {\ ell}}$

If the system is in a magnetic field, the vectors (total angular momentum) and (magnetic moment), whose expected values ​​are always parallel or antiparallel to each other , precess with the Larmor frequency around the direction of the magnetic field and cause an observable splitting of the energy level through which the g-factor can be determined (see Zeeman effect , nuclear magnetic resonance , electron spin resonance ). Such measurements made a decisive contribution to the discovery of the spin and to the explanation of the structure of the electron shell and atomic nuclei. ${\ displaystyle {\ vec {J}}}$${\ displaystyle {\ vec {\ mu}}}$

## theory

### Magnetic moment

According to classical physics, a body with mass and electrical charge that describes a circular path with angular momentum has a magnetic moment : ${\ displaystyle m}$${\ displaystyle q}$${\ displaystyle {\ vec {\ ell}}}$

${\ displaystyle {\ vec {\ mu}} _ {\ ell} = {\ frac {q} {2m}} {\ vec {\ ell}} \ quad}$
(small letters for single-particle systems, large letters for multi-particle systems)

The gyromagnetic ratio according to classical physics is accordingly . If the charge is positive, the magnetic moment is parallel to the angular momentum; if the charge is negative, it is anti-parallel. ${\ displaystyle {\ tfrac {| {\ vec {\ mu}} _ {\ ell} |} {| {\ vec {\ ell}} |}} = {\ tfrac {| q |} {2m}}}$

This also applies to the orbital angular momentum in quantum mechanics (whereby, strictly speaking, the operators or are meant here). On the other hand, the magnetic moment associated with spin (which does not exist in classical physics) deviates from what is taken into account in the formula with an "anomalous spin g-factor" : ${\ displaystyle {\ vec {\ ell}}}$ ${\ displaystyle {\ hat {\ vec {\ mu}}}}$${\ displaystyle {\ hat {\ vec {\ ell}}}}$ ${\ displaystyle {\ hat {\ vec {s}}} {\ mathord {=}} {\ tfrac {\ hbar} {2}} {\ hat {\ vec {\ sigma}}}}$${\ displaystyle g_ {s}}$

${\ displaystyle {\ vec {\ mu}} _ {s} = g_ {s} {\ frac {q} {2m}} {\ vec {s}}}$

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

${\ displaystyle {\ vec {j}} = {\ vec {\ ell}} + {\ vec {s}} \ quad {\ mbox {or}} \ quad {\ vec {\ mu}} = {\ vec {\ mu _ {\ ell}}} + {\ vec {\ mu _ {s}}} = {\ frac {q} {2m}} ({\ vec {\ ell}} + g_ {s} { \ vec {s}})}$

In order to be able to use these formulas in general and also for neutral particles (like the neutron), although their orbital angular momentum does not generate a magnetic moment because of it , one writes ${\ displaystyle q {\ mathord {=}} 0}$

${\ displaystyle {\ vec {\ mu}} = {\ frac {q} {2m}} (g _ {\ ell} {\ vec {\ ell}} + g_ {s} {\ vec {s}}), }$

wherein for neutral particles and is to be inserted. Here the sign depends on whether the magnetic moment is parallel or anti-parallel to the spin. ${\ displaystyle q {\ mathord {=}} {\ mathord {+}} e}$${\ displaystyle g _ {\ ell} {\ mathord {=}} 0}$

If the system consists of several particles of the same kind (e.g. electrons in the atomic shell), then all orbital angular momentum operators add up to the total orbital angular momentum and all spin operators add up to the total spin . The operators for the total angular momentum of the system and its total magnetic moment are the vector sums ${\ displaystyle {\ vec {L}} = \ sum {\ vec {\ ell _ {i}}}}$${\ displaystyle {\ vec {S}} = \ sum {\ vec {s_ {i}}}}$

${\ displaystyle {\ vec {J}} = {\ vec {L}} + {\ vec {S}} \ quad {\ mbox {or}} \ quad {\ vec {\ mu}} = {\ vec {\ mu _ {L}}} + {\ vec {\ mu _ {S}}} = {\ frac {q} {2m}} (g _ {\ ell} {\ vec {L}} + g_ {s } {\ vec {S}}).}$

### Landé formula

Because , the above-defined vectors and not parallel. However, if the quantum numbers for the amounts of orbital, spin and total angular momentum (and its component) have certain values in a state determined by the energy , the magnetic moment only has an effect through its component parallel to ( Wigner-Eckart theorem , illustrated by "The components perpendicular to the angular momentum average out."). To distinguish them from the quantum numbers, the operators are now written "with a roof" (etc.). It should also be noted that the symbols here denote the physical quantities with their dimensions, but the symbols denote the pure quantum numbers. So the operator has e.g. B. the eigenvalue . ${\ displaystyle g _ {\ ell} \ neq g_ {s}}$${\ displaystyle {\ vec {\ jmath}}}$${\ displaystyle {\ vec {\ mu}}}$${\ displaystyle \ ell, s, j, m_ {j}}$${\ displaystyle z}$${\ displaystyle {\ vec {\ jmath}}}$${\ displaystyle {\ hat {\ vec {\ jmath}}}}$${\ displaystyle {\ hat {\ vec {\ jmath}}}, \ {\ hat {\ vec {\ ell}}}, \ {\ hat {\ vec {s}}}, \ {\ hat {\ vec {\ mu}}}}$${\ displaystyle j, \ ell, s}$${\ displaystyle {\ hat {\ vec {\ jmath}}} ^ {2}}$${\ displaystyle j (j + 1) \ hbar ^ {2}}$

So the operator is not a multiple of , but it is effectively replaced by the parallel component , which is also used to determine the resulting factor : ${\ displaystyle {\ hat {\ vec {\ mu}}}}$${\ displaystyle {\ hat {\ vec {\ jmath}}}}$${\ displaystyle {\ hat {\ vec {\ jmath}}}}$${\ displaystyle {\ hat {\ vec {\ mu}}} _ {\ text {eff}}}$${\ displaystyle g}$${\ displaystyle g_ {j}}$

${\ displaystyle {\ hat {\ vec {\ mu}}} _ {\ text {eff}} = {\ frac {({\ hat {\ vec {\ mu}}} \ cdot {\ hat {\ vec { \ jmath}}})} {{\ hat {\ vec {\ jmath}}} ^ {2}}} \, {\ hat {\ vec {\ jmath}}} = g_ {j} {\ frac {q } {2m}} {\ hat {\ vec {\ jmath}}}}$

The resulting factor is therefore the value of the operator ${\ displaystyle g}$

${\ displaystyle g_ {j} = {\ frac {2m} {q}} {\ frac {({\ hat {\ vec {\ mu}}} \ cdot {\ hat {\ vec {\ jmath}}}) } {{\ hat {\ vec {\ jmath}}} ^ {2}}} \ quad = {\ frac {g _ {\ ell} ({\ hat {\ vec {\ ell}}} \ cdot {\ hat {\ vec {\ jmath}}}) + g_ {s} ({\ hat {\ vec {s}}} \ cdot {\ hat {\ vec {\ jmath}}})} {\ hbar ^ {2} j (j + 1)}}.}$

If one squares the first scalar product according to ${\ displaystyle {\ hat {\ vec {s}}} = {\ hat {\ vec {\ jmath}}} - {\ hat {\ vec {\ ell}}}}$

${\ displaystyle ({\ hat {\ vec {\ jmath}}} \ cdot {\ hat {\ vec {\ ell}}}) = {\ tfrac {1} {2}} \, ({\ hat {\ vec {\ jmath}}} ^ {2} + {\ hat {\ vec {\ ell}}} ^ {2} - {\ hat {\ vec {s}}} ^ {2}) = {\ tfrac { 1} {2}} (j (j + 1) + \ ell (\ ell +1) -s (s + 1)) \, \ hbar ^ {2}}$

express by the quantum numbers, the second analogously. The generalized Landé formula follows

${\ displaystyle g_ {j} = {\ frac {1} {2}} {\ frac {g _ {\ ell} (j (j + 1) + \ ell (\ ell +1) -s (s + 1) ) + g_ {s} (j (j + 1) + s (s + 1) - \ ell (\ ell +1))} {j (j + 1)}}.}$

For an electron one sets and and thus obtains the usual Landé formula${\ displaystyle g _ {\ ell} = 1}$${\ displaystyle g_ {s} \ approx 2}$

${\ displaystyle g_ {j} = 1 + {\ frac {\, j (j + 1) - \ ell (\ ell +1) + s (s + 1)} {2j (j + 1)}}.}$

For an entire atomic shell with several electrons, the type of coupling of the angular momentum must be taken into account. The simple Landé formula is correct in the case of the LS coupling , because only then do the total orbital angular momentum and the total spindle angular momentum have well-defined values in the state under consideration . For the calculation, only the valence electrons are taken into account, which are distributed according to Hund's rules to the different levels of the highest occupied shell, since the angular momentum and spin quantum numbers of closed shells couple to zero. ${\ displaystyle {\ hat {\ vec {L}}}}$${\ displaystyle {\ hat {\ vec {S}}}}$

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

Landé himself had stated (almost correctly) in 1923 . Only after the quantum mechanical formulas for angular momentum were discovered in 1925 did the correct version emerge. ${\ displaystyle g = 1 + {\ tfrac {j ^ {2} - {\ tfrac {1} {4}} - \ ell ^ {2} + s ^ {2}} {2 (j ^ {2} - {\ tfrac {1} {4}})}}}$

## Abnormal spin g-factors

### electron

In the theoretical description of the electron using the Schrödinger equation, there is initially no spin. With the discovery of the half-integer spin, the anomalous gyromagnetic factor had to be ascribed to the electron due to the observations of the anomalous Zeeman effect . In the extension of the Schrödinger equation to the Pauli equation , the spin is included, whereby the gyromagnetic factor is freely selectable, i.e. unexplained. Only the relativistic description of the electron by the Dirac equation for spin-½- fermions resulted theoretically . Contrary to popular opinion, this value can also be based on the non-relativistic Schrödinger equation if it is modified appropriately. However, the necessary modifications are not motivated without relativity theory, which is why the widespread view must be allowed to correctly identify the physical causes for the modified g-factor. ${\ displaystyle g_ {e} {\ mathord {=}} 2}$${\ displaystyle g_ {e} {\ mathord {=}} 2}$

Electron spin resonance experiments later showed slight deviations. Sometimes only these additional deviations are called anomalous magnetic moment ; experiments to determine them are also called (g-2) experiments . The Dirac equation does not take into account the possible generation and annihilation of photons and electron- positron pairs. This is what quantum electrodynamics does . This leads to corrections in the coupling of the electron to the magnetic field. They provide a theoretical value of

${\ displaystyle g _ {\, {\ text {Electron, theoretical}}} = 2 {,} 002 \, 319 \, 304 \, 8 (8),}$

whereas experiments according to the current measurement accuracy have a value of

${\ displaystyle g _ {\, {\ text {Electron, measured}}} = 2 {,} 002 \, 319 \, 304 \, 362 \, 56 (35)}$

surrender. The experimental accuracy exceeds the accuracy of the theoretical prediction. The precise calculation of the g-factor and the comparison with the experiment, for example with the muon, is used for precision tests of the standard model of elementary particles.

### Compound particles

Compound particles have significantly different gyromagnetic factors:

${\ displaystyle g _ {\, {\ text {Proton}}} = 5 {,} 585 \, 694 \, 6893 (16) \, \}$
${\ displaystyle g _ {\, {\ text {Neutron}}} = - 3 {,} 826 \, 085 \, 45 (90) \. \}$

The g-factors of these nucleons cannot be calculated precisely because the behavior of their components, quarks and gluons , is not known with sufficient accuracy.

Strictly speaking, the gyromagnetic factor of the neutron is 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.

Just like the gyromagnetic factors of protons and neutrons, the nuclear g-factor cannot be calculated a priori, but has to be determined experimentally.

### Determination history

The g-factor, especially the value for the electron, was introduced phenomenologically by Landé in 1923 in order to express the observations of the anomalous Zeeman effect in formulas. A theoretical explanation was found in 1928 with the Dirac equation . The strongly differing values ​​for proton (1933) and neutron (1948) could only be understood in the quark model decades later . The small deviation from the Dirac value for the electron was discovered in bound electrons by Polkarp Kusch and others from 1946, for free electrons by H. Richard Crane from 1954, up to a precision measurement to 13 decimal places in 2011 by D. Hanneke, in accordance with the theoretical calculation in the standard model of elementary particles. ${\ displaystyle g_ {s} = 2}$${\ displaystyle g_ {s} = 2}$

Vernon Hughes in particular devoted himself to determining the g-factor of the muon , culminating in an experiment at Brookhaven National Laboratory , the results of which were presented in 2002. The comparison with the theory is more difficult for the muon insofar as additional experimental values ​​with less accuracy flow into the theoretical value. An analysis in 2009 revealed a deviation from the predictions of the Standard Model, the Muon g-2 experiment at Fermilab will measure the value more precisely.

## Remarks

1. Two possible approaches:
• According to Shankar: Principles of quantum mechanics. Plenum Press, NY 1980: The kinetic energy held by being of the same size . After coupling the electromagnetic field in the usual form,, we get g = 2.${\ displaystyle {\ hat {\ vec {p}}} ^ {2} / 2m}$${\ displaystyle ({\ vec {\ sigma}} \ cdot {\ hat {\ vec {p}}}) ^ {2} / 2m}$${\ displaystyle {\ hat {\ vec {p}}} \ rightarrow ({\ hat {\ vec {p}}} - {\ tfrac {e} {c}} {\ vec {A}})}$
• According to Greiner: quantum mechanics. Introduction. Volume 4, Chapter XIV, ISBN 3-8171-1765-5 : Linearize the Schrödinger equation directly, d. H. as the product of two differential operators of the first order. The result is the 4-component Dirac spinors and accordingly g = 2.
2. Why, for example, should one factor the linear operator before the wave function in the Schrödinger equation and make an equation out of it in which the wave function has to be replaced by vectors of dimension 4? And why should one expect that the spinor equation that arises after minimal coupling to the electromagnetic field, whose solutions are no longer solutions of the Schrödinger equation obtained analogously , describes the quantum mechanics of the particle in the field more correctly than the Schrödinger equation itself? In the relativistic case, where the Schrödinger equation corresponds to the Klein-Gordon equation and the spinor equation corresponds to the Dirac equation, on the one hand the motivation is clear (relativistic symmetry between time and place and a first-order equation in time are desired), on the other hand, the Klein -Gordon's equation from the outset not seen as the correct description because of the difficulty of defining a probability density.
3. to the QED corrections are ${\ textstyle {\ frac {g-2} {2}}}$
• in : See ME Peskin, Schroeder DV: An Introduction to Quantum Field Theory , Addison-Wesley (1995), Chapter 6: Radiative Corrections . There J.Schwinger, Phys Rev. 73, 416L (1948) is given as the source.${\ textstyle {\ cal {O (\ alpha)}}}$${\ textstyle {\ frac {\ alpha} {2 \ pi}} = 0.0011614 ...}$
• in the fourth order, including : See JJ Sakurai: Advanced Quantum Mechanics , Addison-Wesley (1967), chapter 4-7: Mass and Charge Renormalization; Radiative Corrections . There, C. Sommerfield and A. Petersen are cited (presumably CM Sommerfield, Phys Rev. 107, 328 (1957); Ann. Phys. (NY) 5, 26 (1958); and A. Petermann, Helv Phys. Acta 30, 407 (1957); Nucl. Phys. 3, 689 (1957)).${\ textstyle {\ cal {O (\ alpha ^ {2})}}}$${\ textstyle {\ frac {\ alpha} {2 \ pi}} - 0.328 ({\ frac {\ alpha} {2 \ pi}}) ^ {2} = 0.0011596 ...}$
• in the sixth order, so : here in the eighth decimal place there is another term that, according to Stanley J. Brodsky and Ralph Roskies: Quantum Electrodynamics and Renormalization Theory in the Infinite Momentum Frame , SLAC-PUB-1100 (TH) and (EXP) , August 1972, inspirehep.net (PDF) is attached , which agrees with von Levine and Wright, who compared M. Levine and J. Wright, Phys. Rev. Letters, 26: 1351 (1971); Proceedings of the Second Colloquium on Advanced Computing Methods in Theoretical Physics, Marseille (1971), and private commumcation .${\ textstyle {\ cal {O (\ alpha ^ {3})}}}$${\ textstyle 2 (1.11 \ pm 0.23) ({\ frac {\ alpha} {\ pi}}) ^ {3}}$${\ textstyle 2 (0.90 \ pm 0.02) ({\ frac {\ alpha} {\ pi}}) ^ {3}}$

## literature

• Jörn Bleck-Neuhaus: Elementary Particles. Modern physics from the atoms to the standard model . Springer, Heidelberg 2010, ISBN 978-3-540-85299-5 .

## Individual evidence

1. A. Landé: Term structure and Zeeman effect of the multiplets. In: Journal of Physics. Volume 15, pp. 189-205 (1923) doi: 10.1007 / BF01330473 .
2. CODATA Recommended Values. National Institute of Standards and Technology, accessed July 20, 2019 . Value for . The numbers in brackets denote the uncertainty in the last digits of the value; this uncertainty is given as the estimated standard deviation of the specified numerical value from the actual value. The value for CODATA uses the opposite sign convention.${\ displaystyle g _ {\ text {electron}}}$
3. CODATA Recommended Values. National Institute of Standards and Technology, accessed July 20, 2019 . Value for . The numbers in brackets denote the uncertainty in the last digits of the value; this uncertainty is given as the estimated standard deviation of the specified numerical value from the actual value.${\ displaystyle g _ {\, {\ text {Proton}}}}$
4. CODATA Recommended Values. National Institute of Standards and Technology, accessed July 20, 2019 . Value for . The numbers in brackets denote the uncertainty in the last digits of the value; this uncertainty is given as the estimated standard deviation of the specified numerical value from the actual value.${\ displaystyle g _ {\, {\ text {Neutron}}}}$
5. D. Hanneke, p Fogwell Hoogerheide, G. Gabrielse: Cavity control of a single-electron quantum cyclotron: Measuring the electron magnetic moment. In: Physical Review A , Volume 83, 2011, p. 052122, doi: 10.1103 / PhysRevA.83.052122 .
6. ^ Final Report. Brookhaven, Physical Review D, Volume 73, 2006, arxiv : hep-ex / 0602035 .
7. ^ Fred Jegerlehner, Andreas Nyffeler: The muon g − 2. In: Physics Reports. Volume 477, 2009, pp. 1-111.