Gyromagnetic ratio: Difference between revisions
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{{cite web |title=Nuclear magneton|work=Fundamental physical constants |url=http://physics.nist.gov/cgi-bin/cuu/Value?eqmun|search_for=muN |publisher=NIST |accessdate=2011-03-28}} | {{cite web |title=Nuclear magneton|work=Fundamental physical constants |url=http://physics.nist.gov/cgi-bin/cuu/Value?eqmun|search_for=muN |publisher=NIST |accessdate=2011-03-28}} | ||
</ref> | </ref> | ||
:<math> \mu_{\rm N} = \frac{e\hbar}{2m_{\rm p}} \ , </math> | :<math> \mu_{\rm N} = \frac{e\hbar}{2m_{\rm p}} =\mu_e \frac{m_e}{m_p} =\mathrm{5.050\ 783\ 24\ \times \ 10^{-27}\ J\ T^{-1}} \ , </math> | ||
with ''m<sub>p</sub>'' the mass of the proton, and the proton ''g''-factor is:<ref name=NIST7> | with ''m<sub>p</sub>'' the mass of the proton, and the proton ''g''-factor is:<ref name=NIST7> | ||
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</ref> | </ref> | ||
:<math>g_{\rm p} = 2\mu_{\rm p} | :<math>g_{\rm p} = 2\frac{\mu_{\rm p}}{\mu_{\rm N}} = \mathrm{5.585 694 713} \ , </math> | ||
and this surprising value suggests the proton is not a simple particle, but a complex structure. | |||
==Notes== | ==Notes== | ||
<references/> | <references/> | ||
Revision as of 17:53, 28 March 2011
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:
Its 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.
Examples
The electron gyromagnetic ratio is:[1]
where μe is the magnetic moment of the electron (-928.476 377 x 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:[2]
where μp is the magnetic moment of the proton (1.410 606 662 x 10-26 J T-1). Other ratios can be found on the NIST web site.[3]
Theory and experiment
The relativistic quantum mechanical theory provided by the Dirac equation predicted the electron to have a magnetic moment of exactly one Bohr mangeton, where the Bohr magneton is:[4]
with e the elementary charge. If magnetic moment is expressed in units of Bohr magnetons, taking the spin 1/2 of the electron into account, the gyromagnetic ratio becomes the g-factor:[5]
The Dirac prediction μe = μB results in a g-factor of exactly ge = 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.[6]
Similarly, the nuclear magneton is defined:[7]
with mp the mass of the proton, and the proton g-factor is:[8]
and this surprising value suggests the proton is not a simple particle, but a complex structure.
Notes
- ↑ Electron gyromagnetic ratio. The NIST reference on constants, units, and uncertainty. Retrieved on 2011-03-28.
- ↑ Proton gyromagnetic ratio. The NIST reference on constants, units, and uncertainty. Retrieved on 2011-03-28.
- ↑ A general search menu for the NIST database is found at CODATA recommended values for the fundamental constants. National Institute of Standards and Technology. Retrieved on 2011-03-28.
- ↑ Bohr magneton. Fundamental physical constants. NIST. Retrieved on 2011-03-28.
- ↑ Electron g factor. Fundamental physical constants. NIST. Retrieved on 2011-03-28.
- ↑ An historical summary can be found in Toichiro Kinoshita (2010). “§3.2.2 Early tests of QED”, B. Lee Roberts, William J. Marciano, eds: Lepton dipole moments. World Scientific, pp. 73 ff. ISBN 9814271837. An introduction to the behavior of the electron in a magnetic flux is found in Yehuda Benzion Band (2006). “§5.1.1 Electron spin coupling”, Light and matter: electromagnetism, optics, spectroscopy and lasers. Wiley, pp. 297 ff. ISBN 0471899313.
- ↑ Nuclear magneton. Fundamental physical constants. NIST. Retrieved on 2011-03-28.
- ↑ Proton g factor. Fundamental physical constants. NIST. Retrieved on 2011-03-28.