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

• for the free proton :

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

• for the electron :

${\displaystyle \gamma _{\text{Elektron}}=1{,}760\,859\,630\,23(53)\cdot 10^{11}\ \mathrm {s} ^{-1}\,\mathrm {T} ^{-1}\,}$

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

See also

• Larmor frequency

literature

• Horst Stöcker : Pocket book of physics. 4th edition, Verlag Harry Deutsch, Frankfurt am Main, 2000, ISBN 3-8171-1628-4 .
• Hermann hook , Hans Christoph Wolf : atomic and quantum physics . 8th edition, Springer-Verlag, Berlin Heidelberg New York, 2004, pp. 194 ff, ISBN 3-540-02621-5 .

Individual evidence

1. ↑ Manfred Hesse, Herbert Meier, Bernd Zeeh: Spectroscopic methods in organic chemistry . 7th edition, Georg Thieme Verlag, Stuttgart, 2005, ISBN 3-13-576107-X
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. ${\displaystyle \gamma _{e}}$
4. ↑ MA Bernstein, KF King and XJ Zhou: Handbook of MRI Pulse Sequences . Elsevier Academic Press, San Diego 2004, ISBN 0-12-092861-2 , p. 960.
5. ↑ RC Weast, MJ Astle (Ed.): Handbook of Chemistry and Physics . CRC Press, Boca Raton 1982, ISBN 0-8493-0463-6 , p. E66.
6. ↑ proton gyromagnetic ratio . NIST . 2019.
7. ↑ proton gyromagnetic ratio over 2 pi . NIST . 2019.