Gyromagnetic ratio

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The gyromagnetic ratio (sometimes magnetogyric ratio), γ, is the constant of proportionality between the magnetic moment (μ) and the angular momentum(J) of an object:

${\displaystyle \mathit{\mu }=\pm \gamma \mathbf{𝐉},}$

where the sign is chosen to make γ a positive number.

The units of the gyromagnetic ratio are SI units are radian per second per tesla (s⁻¹·T⁻¹) or, equivalently, coulomb per kilogram (C·kg⁻¹). When the object is placed in a magnetic flux density B, because of its magnetic moment it experiences a torque and precesses about the field at the Larmor frequency, which is given (in radians/s) by the product of the field strength and the gyromagnetic ratio.^[1]

A closely related quantity is the g-factor, which relates the magnetic moment in units of magnetons to spin: in terms of the gyromagnetic ratio, g = ±γℏ /μ with ℏ the reduced Planck constant and μ the appropriate magneton (the Bohr magneton for electrons and the nuclear magneton for nucleii). The sign of the g-factor is negative when the magnetic moment is oriented opposite to the angular momentum (it is negative for electrons and neutrons) and positive when the two are aligned the same way (it is positive for protons). More detail is below.

Examples[edit]

The electron gyromagnetic ratio is:^[2]

${\displaystyle {\gamma }_{\mathrm{e}}=2|{\mu }_{\mathrm{e}}|/\hslash =\mathrm{1}\mathrm{.}\mathrm{7}\mathrm{6}\mathrm{0}\mathrm{8}\mathrm{5}\mathrm{9}\mathrm{7}\mathrm{7}\mathrm{0}\mathrm{\times }\mathrm{1}{\mathrm{0}}^{\mathrm{1}\mathrm{1}}{\mathrm{s}}^{\mathrm{-}\mathrm{1}}{\mathrm{T}}^{\mathrm{-}\mathrm{1}},}$

where μ_e is the magnetic moment of the electron (−928.476 377 × 10⁻²⁶ J T⁻¹), and ℏ is Planck's constant divided by 2π and ℏ/2 is the spin angular momentum.

Similarly, the proton gyromagnetic ratio is:^[3]

${\displaystyle {\gamma }_{\mathrm{p}}=2{\mu }_{\mathrm{p}}/\hslash =\mathrm{2}\mathrm{.}\mathrm{6}\mathrm{7}\mathrm{5}\mathrm{2}\mathrm{2}\mathrm{2}\mathrm{0}\mathrm{9}\mathrm{9}\mathrm{\times }\mathrm{1}{\mathrm{0}}^{\mathrm{8}}{\mathrm{s}}^{\mathrm{-}\mathrm{1}}{\mathrm{T}}^{\mathrm{-}\mathrm{1}},}$

where μ_p is the magnetic moment of the proton (1.410 606 662 × 10⁻²⁶ J T⁻¹).

The neutron gyromagnetic ratio is:^[4]

${\displaystyle {\gamma }_{\mathrm{n}}=2|{\mu }_{\mathrm{n}}|/\hslash =\mathrm{1}\mathrm{.}\mathrm{8}\mathrm{3}\mathrm{2}\mathrm{4}\mathrm{7}\mathrm{1}\mathrm{8}\mathrm{5}\mathrm{\times }\mathrm{1}{\mathrm{0}}^{\mathrm{8}}{\mathrm{s}}^{\mathrm{-}\mathrm{1}}{\mathrm{T}}^{\mathrm{-}\mathrm{1}},}$

where μ_n is the magnetic moment of the neutron (−0.966 236 41 × 10⁻²⁶ J T⁻¹).

Other ratios can be found on the NIST web site.^[5]

Theory and experiment; g-factors[edit]

Comparison between theory and experiment for particles usually is made using the g-factor rather than the gyromagnetic ratio because it is a dimensionless number.

Electron[edit]

The relativistic quantum mechanical theory provided by the Dirac equation predicted the electron to have a magnetic moment of exactly one Bohr magneton, where the Bohr magneton is:^[6]

${\displaystyle {\mu }_B=\frac{e\hslash }{2m_e}=\mathrm{9}\mathrm{2}\mathrm{7}\mathrm{.}\mathrm{4}\mathrm{0}\mathrm{0}\mathrm{9}\mathrm{1}\mathrm{5}\mathrm{\times }\mathrm{1}{\mathrm{0}}^{\mathrm{-}\mathrm{2}\mathrm{6}}\mathrm{J}/\mathrm{T},}$

with e the elementary charge. If magnetic moment is expressed in units of Bohr magnetons, the gyromagnetic ratio becomes the g-factor and the magnetic moment becomes:

${\displaystyle {\mu }_e=-{\gamma }_e\frac{\hslash }{2}=\frac{{\mu }_e}{{\mu }_B}{\mu }_B=g_e\frac{{\mu }_B}{2},}$

so the gyromagnetic ratio and the g-factor are related as:

${\displaystyle g_e=-{\gamma }_e\frac{\hslash }{{\mu }_B}.}$

The value of the g-factor for the electron is:^[7]

${\displaystyle g_e=2\frac{{\mu }_e}{{\mu }_B}=\mathrm{-}\mathrm{2}\mathrm{.}\mathrm{0}\mathrm{0}\mathrm{2}\mathrm{3}\mathrm{1}\mathrm{9}\mathrm{3}\mathrm{0}\mathrm{4}\mathrm{3}\mathrm{6}\mathrm{2}\mathrm{2},}$

The Dirac prediction μ_e = μ_B results in a g-factor of exactly g_e = −2. Subsequently (in 1947) experiments on the Zeeman splitting of the gallium atom in magnetic field showed that was not exactly the case, and later this departure was calculated using quantum electrodynamics.^[8]

Proton[edit]

Similarly, the nuclear magneton is defined as:^[9]

${\displaystyle {\mu }_{\mathrm{N}}=\frac{e\hslash }{2m_{\mathrm{p}}}={\mu }_B\frac{m_e}{m_p}=\mathrm{5}\mathrm{.}\mathrm{0}\mathrm{5}\mathrm{0}\mathrm{7}\mathrm{8}\mathrm{3}\mathrm{2}\mathrm{4}\mathrm{\times }\mathrm{1}{\mathrm{0}}^{\mathrm{-}\mathrm{2}\mathrm{7}}\mathrm{J}{\mathrm{T}}^{\mathrm{-}\mathrm{1}},}$

with m_p the mass of the proton, and the proton g-factor is:^[10]

${\displaystyle g_{\mathrm{p}}=2\frac{{\mu }_{\mathrm{p}}}{{\mu }_{\mathrm{N}}}=\mathrm{5}\mathrm{.}\mathrm{5}\mathrm{8}\mathrm{5}\mathrm{6}\mathrm{9}\mathrm{4}\mathrm{7}\mathrm{1}\mathrm{3},}$

corresponding to a proton magnetic moment of about μ_p = 2.793 nuclear magnetons.

This surprising value suggests the proton is not a simple particle, but a complex structure, for example, an assembly of quarks. So far, a theoretical calculation of the magnetic moment of the proton in terms of quarks exchanging gluons is a work in progress, with the present estimate as 2.73 nuclear magnetons.^[11]

Neutron[edit]

The neutron g-factor is:^[12]

${\displaystyle g_{\mathrm{n}}=2{\mu }_{\mathrm{n}}/(e\hslash /2m_{\mathrm{p}})=\mathrm{-}\mathrm{3}\mathrm{.}\mathrm{8}\mathrm{2}\mathrm{6}\mathrm{0}\mathrm{8}\mathrm{5}\mathrm{4}\mathrm{5},}$

corresponding to a neutron magnetic moment of about μ_n = −1.913 nuclear magnetons. The theoretical calculation of the magnetic moment of the neutron in terms of quarks exchanging gluons is −1.82 nuclear magnetons.^[11]

Deuteron[edit]

The deuteron is a bound system consisting of a neutron and proton. Because both constituents are spin 1/2 particles, the bound state must have both spins parallel.^[13] The magnetic moment for the deuteron in nuclear magnetons is:^[14]

${\displaystyle {\mu }_{\mathrm{d}}/{\mu }_{\mathrm{N}}={\mu }_{\mathrm{d}}/(e\hslash /2m_{\mathrm{p}})=\mathrm{0}\mathrm{.}\mathrm{8}\mathrm{5}\mathrm{7}\mathrm{4}\mathrm{3}\mathrm{8}\mathrm{2}\mathrm{3}\mathrm{0}\mathrm{8},}$

the same as its g-factor because it is a spin 1 particle. The neutron and proton magnetic moments oppose each other, with a vector sum of about 0.880 nuclear magnetons.

Measurement[edit]

The basis for measuring gyromagnetic ratios is the relation between the spin precession frequency ω of the object (the Larmor frequency) and the magnetic flux density B:

${\displaystyle \omega =\gamma |\mathbf{𝐁}|.}$

The resonance frequency is affected by the surrounding medium and, for example, for protons in water the resulting values are called "shielded" values, referring to the shielding by the electrons in the water molecule, and are denoted with a prime: γ'_p.^[15] This dependence of the resonance frequency upon the environment of a nucleus is called a "chemical shift" and used to explore the matrix surrounding the nucleus in the field of nuclear magnetic resonance (NMR).^[16]

Notes[edit]