# Gyromagnetic ratio

The gyromagnetic ratio (also: magnetogyric ratio ) describes the proportionality factor between the angular momentum (or spin) of a particle and the associated magnetic moment${\ displaystyle \ gamma}$${\ displaystyle {\ vec {X}}}$ ${\ displaystyle {\ vec {\ mu}} _ {X}}$

${\ displaystyle {\ vec {\ mu}} _ {X} = \ gamma _ {X} {\ vec {X}}}$.

Therefore follows: . The internationally used unit of the gyromagnetic ratio is A · s · kg −1 or s −1 · T −1 . ${\ displaystyle \ gamma _ {X} = {\ frac {| {\ vec {\ mu}} _ {X} |} {| {\ vec {X}} |}}}$

The gyromagnetic ratio of a charged particle is the product of its ( dimensionless ) gyromagnetic factor and its magneton , based on the reduced Planck quantum : ${\ displaystyle g}$ ${\ displaystyle \ mu}$ ${\ displaystyle \ hbar}$

${\ displaystyle \ gamma = g \, {\ frac {\ mu} {\ hbar}}}$

With

• ${\ displaystyle \ mu = {\ frac {q} {2 \, m}} \, \ hbar}$the magneton of the particle
• ${\ displaystyle q}$: electric charge
• ${\ displaystyle m}$: Particle mass.

The gyromagnetic ratio can be determined using the Barnett effect and the Einstein-de-Haas effect . In many other experiments, such as B. ferromagnetic resonance or electron spin resonance , the value can deviate significantly from - in this case one speaks of the spectroscopic splitting factor or ratio . ${\ displaystyle \ gamma}$

## γ ℓ for pure orbital angular momentum of an electron

As explained in the article Magnetic Moment , the following applies to the magnetic moment of the orbital angular momentum of an electron:

${\ displaystyle {\ vec {\ mu _ {\ ell}}} = - {\ frac {e} {2m_ {e}}} {\ vec {\ ell}}}$.

With

• ${\ displaystyle -e}$ the charge of the electron
• ${\ displaystyle m_ {e}}$ its mass.

Hence it follows:

${\ displaystyle \ gamma _ {\ ell} = {\ frac {| {\ vec {\ mu _ {\ ell}}} |} {| {\ vec {\ ell}} |}} = {\ frac {e } {2m_ {e}}} = {\ frac {g _ {\ ell} \ mu _ {B}} {\ hbar}}}$

With

• ${\ displaystyle \ mu _ {B}}$the Bohr magneton . So the g-factor for the orbital movement is${\ displaystyle g _ {\ ell} = 1.}$

## γ S for the spin of a particle

If one considers a particle with spin , then the following applies: ${\ displaystyle {\ vec {S}}}$

${\ displaystyle {\ vec {\ mu}} _ {S} = \ gamma _ {S} {\ vec {S}}}$, respectively ${\ displaystyle \ gamma _ {S} = {\ frac {| {\ vec {\ mu _ {S}}} |} {| {\ vec {S}} |}}}$

The value of this natural constant is characteristic for every type of particle. According to the current measurement accuracy , it is

${\ displaystyle \ gamma _ {\ text {Proton}} = 2 {,} 675 \, 221 \, 8744 (11) \ cdot 10 ^ {8} \ \ mathrm {s} ^ {- 1} \, \ mathrm {T} ^ {- 1} \,}$
${\ displaystyle \ gamma _ {\ text {electron}} = 1 {,} 760 \, 859 \, 630 \, 23 (53) \ cdot 10 ^ {11} \ \ mathrm {s} ^ {- 1} \ , \ mathrm {T} ^ {- 1} \,}$

the numbers in brackets indicate the estimated standard deviation for the mean value , which corresponds to the last two numbers before the brackets.

The g-factor for spin magnetism for the free electron is almost exactly - with the exception of seven places behind the decimal point - equal to 2. For the free proton, however, the same is by no means true: the magnetic moment of the proton is of the order of the so-called " nuclear magneton " ( that would be the value ), but it is an odd multiple of this value, more precisely: 2.79 times. The neutron also has a magnetic moment, although as a whole it is electrically neutral. Its magnetic moment is −1.91 times that of the nuclear magneton and thus points opposite to that of the proton. It can be explained by the substructure of the neutron. ${\ displaystyle | e | \ hbar / (2m _ {\ mathrm {Proton}}) \,}$

The ferromagnetic metals iron, cobalt and nickel have electronic g-factors pretty close to 2 (e.g. only about 10% more or less); This means that the magnetism of these systems is predominantly spin magnetism, but with a small orbital component.

## Gyromagnetic relationships of atomic nuclei

This ratio can also be measured and specified for cores. Some values ​​are given in the following table.

core ${\ displaystyle \ gamma _ {n}}$
in 10 7 rad · s −1 · T −1
${\ displaystyle \ gamma _ {n} / 2 \ pi}$
in MHz · T −1
1 H. +26.752 +42,577
2 H 0+4.1065 0+6.536
3 He −20.3789 −32.434
7 li +10.3962 +16,546
13 C 0+6.7262 +10.705
14 N. 0+1.9331 0+3.077
15 N. 0−2.7116 0−4.316
17 O 0−3.6264 0−5.772
19 F +25.1662 +40.053
23 Well 0+7.0761 +11.262
31 P. +10.8291 +17.235
129 Xe 0−7.3997 −11.777

2. CODATA Recommended Values. National Institute of Standards and Technology, accessed July 16, 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.${\ displaystyle \ gamma _ {p}}$
3. CODATA Recommended Values. National Institute of Standards and Technology, accessed July 16, 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.${\ displaystyle \ gamma _ {e}}$