Gyromagnetic ratio: Difference between revisions
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The '''gyromagnetic ratio''' (sometimes ''magnetogyric ratio''), | 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: | ||
:<math> | :<math> | ||
\boldsymbol {\mu} = \gamma \mathbf{J} \ | \boldsymbol {\mu} = \pm \gamma \mathbf{J} \ ,</math> | ||
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 (unit)|tesla]]''' (s<sup>−1</sup>·T<sup>−1</sup>) or, equivalently, '''[[coulomb]] per [[kilogram]]''' (C·kg<sup>−1</sup>). 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.<ref name = Larmor> | |||
For a number of entertaining exercises on this subject, see {{cite book |title= Advanced Tutorials for the Biomedical Sciences: Animations, Simulations, and Calculations Using Mathematica |author=Charles Pidgeon |chapter=Larmor frequency versus field strength |url=http://books.google.com/books?id=oNXIPJuMefcC&pg=PA165 |pages=pp. 165 ''ff'' |isbn=0471186465 |year=1996 |publisher=Wiley-VCH }} | |||
</ref> | |||
A closely related quantity is the '''''g'''''-'''factor''', which relates the magnetic moment in units of ''magnetons'' to [[Pauli spin matrices|spin]]: in terms of the gyromagnetic ratio, ''g = ±γ''ℏ /''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 [[electron]]s and [[neutrons]]) and positive when the two are aligned the same way (it is positive for [[proton]]s). More detail is below. | |||
==Examples== | ==Examples== | ||
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:<math> \gamma_{\rm e} = 2|\mu_{\rm e}|/\hbar = \mathrm{ 1.760\ 859\ 770\ \times \ 10^{11}\ s^{-1} T^{-1} }\ , </math> | :<math> \gamma_{\rm e} = 2|\mu_{\rm e}|/\hbar = \mathrm{ 1.760\ 859\ 770\ \times \ 10^{11}\ s^{-1} T^{-1} }\ , </math> | ||
where ''μ<sub>e</sub>'' is the [[magnetic moment]] of the electron ( | where ''μ<sub>e</sub>'' is the [[magnetic moment]] of the electron {{nowrap|(−928.476 377 × 10<sup>−26</sup> J T<sup>−1</sup>),}} and ℏ is [[Planck's constant]] divided by 2π and ℏ/2 is the spin angular momentum. | ||
Similarly, the ''proton'' gyromagnetic ratio is:<ref name=NIST1> | Similarly, the ''proton'' gyromagnetic ratio is:<ref name=NIST1> | ||
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</ref> | </ref> | ||
:<math> \gamma_{\rm p} = 2\mu_{\rm p}/\hbar = \mathrm{ 2.675\ 222\ 099\ \times \ 10^{ | :<math> \gamma_{\rm p} = 2\mu_{\rm p}/\hbar = \mathrm{ 2.675\ 222\ 099\ \times \ 10^{8}\ s^{-1} T^{-1} }\ , </math> | ||
where ''μ<sub>p</sub>'' is the [[magnetic moment]] of the proton {{nowrap|(1.410 606 662 × 10<sup>−26</sup> J T<sup>−1</sup>).}} | |||
The ''neutron'' gyromagnetic ratio is:<ref name=NIST1a> | |||
{{cite web |title=Neutron gyromagnetic ratio |url=http://physics.nist.gov/cgi-bin/cuu/Value?eqgamman|search_for=neutron+gyromagnetic+ratio |work=The NIST reference on constants, units, and uncertainty |accessdate=2011-03-28}} | |||
</ref> | |||
:<math> \gamma_{\rm n} = 2|\mu_{\rm n}|/\hbar = \mathrm{ 1.832\ 471\ 85 \times 10^8\ s^{-1} T^{-1} }\ , </math> | |||
where ''μ<sub> | where ''μ<sub>n</sub>'' is the [[magnetic moment]] of the neutron {{nowrap|(−0.966 236 41 × 10<sup>−26</sup> J T<sup>−1</sup>).}} | ||
A general search menu for the NIST database is found at {{cite web |title=CODATA recommended values for the fundamental constants |url=http://physics.nist.gov/cuu/Constants/index.html |accessdate 2011-03-28 |publisher= [[National Institute of Standards and Technology]] |accessdate=2011-03-28}} | Other ratios can be found on the NIST web site.<ref name=NIST2> | ||
A general search menu for the NIST database is found at {{cite web |title=CODATA recommended values for the fundamental constants |url=http://physics.nist.gov/cuu/Constants/index.html |accessdate 2011-03-28 |publisher= [[National Institute of Standards and Technology]] |accessdate=2011-03-28}} For example, the [http://physics.nist.gov/cgi-bin/cuu/Results?search_for=magnetic+moment magnetic moment]. | |||
</ref> | </ref> | ||
==Theory and experiment== | ==Theory and experiment; ''g''-factors== | ||
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=== | ||
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:<ref name=NIST4> | |||
{{cite web |title=Bohr magneton |work=Fundamental physical constants |url=http://physics.nist.gov/cgi-bin/cuu/Value?eqmub|search_for=Bohr+magneton |publisher=NIST |accessdate=2011-03-28}} | |||
Subsequently (in 1947) experiments on the [[Zeeman effect|Zeeman splitting]] of the gallium atom in magnetic field showed that was not exactly the case, and | </ref> | ||
:<math>\mu_B = \frac{e \hbar }{2 m_e } = \mathrm{927.400 915 \times 10^{-26}\ J/T}\ , </math> | |||
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: | |||
:<math>\mu_e = -\gamma_e \frac {\hbar}{2} = \frac{\mu_e}{\mu_B} {\mu_B} = g_e \frac {\mu_B}{2} \ , </math> | |||
so the gyromagnetic ratio and the ''g''-factor are related as: | |||
:<math> g_e = -\gamma_e\frac{ \hbar}{\mu_B} \ . </math> | |||
The value of the ''g''-factor for the electron is:<ref name=NIST5> | |||
{{cite web |title=Electron g factor|work=Fundamental physical constants |url=http://physics.nist.gov/cgi-bin/cuu/Value?eqgem|search_for=g+factor |publisher=NIST |accessdate=2011-03-28}} | |||
</ref> | |||
:<math> g_e = 2\frac{\mu_e}{\mu_B} =\mathrm{ -2.002 319 304 3622 } \ , </math> | |||
The Dirac prediction ''μ<sub>e</sub> = μ<sub>B</sub>'' results in a ''g''-factor of exactly ''g<sub>e</sub>'' = −2. Subsequently (in 1947) experiments on the [[Zeeman effect|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]].<ref name= Marciano> | |||
An historical summary can be found in {{cite book |title=Lepton dipole moments |author=Toichiro Kinoshita|editor=B. Lee Roberts, William J. Marciano, eds |url=http://books.google.com/books?id=1TNXUSxJs6IC&pg=PA73 |pages=pp. 73 ''ff''|chapter=§3.2.2 Early tests of QED |isbn=9814271837 |year=2010 |publisher=World Scientific}} An introduction to the behavior of the electron in a magnetic flux is found in {{cite book |title=Light and matter: electromagnetism, optics, spectroscopy and lasers |url=http://books.google.com/books?id=0Rar9yGfQfgC&pg=PA297 |pages=pp. 297 ''ff'' |chapter=§5.1.1 Electron spin coupling |isbn=0471899313 |publisher=Wiley |year=2006 |author=Yehuda Benzion Band}} | An historical summary can be found in {{cite book |title=Lepton dipole moments |author=Toichiro Kinoshita|editor=B. Lee Roberts, William J. Marciano, eds |url=http://books.google.com/books?id=1TNXUSxJs6IC&pg=PA73 |pages=pp. 73 ''ff''|chapter=§3.2.2 Early tests of QED |isbn=9814271837 |year=2010 |publisher=World Scientific}} An introduction to the behavior of the electron in a magnetic flux is found in {{cite book |title=Light and matter: electromagnetism, optics, spectroscopy and lasers |url=http://books.google.com/books?id=0Rar9yGfQfgC&pg=PA297 |pages=pp. 297 ''ff'' |chapter=§5.1.1 Electron spin coupling |isbn=0471899313 |publisher=Wiley |year=2006 |author=Yehuda Benzion Band}} | ||
</ref> | |||
===Proton=== | |||
Similarly, the ''nuclear magneton'' is defined as:<ref name=NIST6> | |||
{{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> | |||
:<math> \mu_{\rm N} = \frac{e\hbar}{2m_{\rm p}} =\mu_B \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> | |||
{{cite web |title=Proton g factor |work=Fundamental physical constants |url=http://physics.nist.gov/cgi-bin/cuu/Value?eqgp|search_for=proton+g-factor |publisher=NIST |accessdate=2011-03-28}} | |||
</ref> | |||
:<math>g_{\rm p} = 2\frac{\mu_{\rm p}}{\mu_{\rm N}} = \mathrm{5.585 694 713} \ , </math> | |||
corresponding to a proton magnetic moment of about ''μ<sub>p</sub>'' = 2.793 nuclear magnetons. | |||
This surprising value suggests the [[proton]] is not a simple particle, but a complex structure, for example, an assembly of [[quark]]s. So far, a theoretical calculation of the magnetic moment of the proton in terms of quarks exchanging [[gluon]]s is a work in progress, with the present estimate as 2.73 nuclear magnetons.<ref name=gluon> | |||
See, for example, {{cite book |chapter=Table 3.5 |url=http://books.google.com/books?id=ws8QZ2M5OR8C&pg=PA103 |pages=p. 104 |title=Nuclear and Particle Physics: An Introduction |author=Brian Martin |isbn=0470742747 |year=2009 |edition=2nd ed|publisher=Wiley}} and {{cite book |title=The spin structure of the proton |author=Steven D. Bass |chapter=Chapter 1: Introduction |pages=pp. 1 ''ff'' |url=http://books.google.com/books?id=VPYoT4GQUV0C&pg=PA1 |isbn=9812709460 |year=2008 |publisher=World Scientific}} | |||
</ref> | |||
===Neutron=== | |||
The neutron ''g''-factor is:<ref name=NIST8> | |||
{{cite web |title=Neutron g factor |work=Fundamental physical constants |url=http://physics.nist.gov/cgi-bin/cuu/Value?eqgnn|search_for=Neutron+g-factor |publisher=NIST |accessdate=2011-03-28}} | |||
</ref> | |||
:<math>g_{\rm n} = 2\mu_{\rm n}/(e \hbar /2m_{\rm p}) = \mathrm{-3.826 085 45 } \ , </math> | |||
corresponding to a neutron [[magnetic moment]] of about ''μ<sub>n</sub>'' = −1.913 nuclear magnetons. The theoretical calculation of the magnetic moment of the neutron in terms of quarks exchanging [[gluon]]s is −1.82 nuclear magnetons.<ref name=gluon/> | |||
===Deuteron=== | |||
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.<ref name=Patel> | |||
For a discussion of the deuteron, see for example {{cite book |title=Nuclear physics: an introduction |author=S. B. Patel |url=http://books.google.com/books?id=aMzuEby4fX4C&pg=PA276 |pages=pp. 276 ''ff'' |chapter= §8.2 Ground state of the deuteron |isbn=8122401252 |publisher=New Age International |year=1991}} | |||
</ref> The magnetic moment for the deuteron in nuclear magnetons is:<ref name=NIST10> | |||
{{cite web |title=Deuteron magnetic moment to nuclear magneton ratio |work=Fundamental physical constants |url=http://physics.nist.gov/cgi-bin/cuu/Value?eqmudsmun|search_for=deuteron |publisher=NIST |accessdate=2011-03-28}} | |||
</ref> | |||
:<math> \mu_{\rm d}/\mu_{\rm N} = \mu_{\rm d}/(e\hbar/2m_{\rm p}) = \mathrm{ 0.857 438 2308 } \ , </math> | |||
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== | |||
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''''': | |||
:<math>\omega = \gamma |\mathbf B | \ . </math> | |||
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'<sub>p</sub>''.<ref name=Quinn> | |||
For example, see {{cite book |title=Metrologia e costanti fondamentali recenti sviluppi: Volume 146 of Proceedings of the International School of Physics "Enrico Fermi" |url=http://books.google.com/books?id=WE22Fez60EcC&pg=PA128 |pages=pp. 128 ''ff'' |editor=T. J. Quinn, S. Leschiutta, Patrizia Tavella, eds |author=BW Petley |chapter=Chapter 2: The fundamental constants and metrology; §3 The fundamental constants and electrical measurements |isbn=1586031678 |year=2001 |publisher=IOS Press}} | |||
</ref> 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 [[NMR spectroscopy|''nuclear magnetic resonance'']] (NMR).<ref name=Graaf> {{cite book |title=In vivo NMR spectroscopy: principles and techniques |author=Robin A. De Graaf |url= http://books.google.com/books?id=048W3-pd3TEC&pg=PA18 |pages=p. 18 |chapter=Chemical shift|year=2007 |isbn=0470026707 |edition=2nd ed|publisher=John Wiley and Sons}} | |||
</ref> | </ref> | ||
==Notes== | ==Notes== | ||
{{reflist|2}}[[Category:Suggestion Bot Tag]] |
Latest revision as of 17:00, 24 August 2024
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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:
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
The electron gyromagnetic ratio is:[2]
where μe 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]
where μp is the magnetic moment of the proton (1.410 606 662 × 10−26 J T−1).
The neutron gyromagnetic ratio is:[4]
where μn 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
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
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]
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:
so the gyromagnetic ratio and the g-factor are related as:
The value of the g-factor for the electron is:[7]
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.[8]
Proton
Similarly, the nuclear magneton is defined as:[9]
with mp the mass of the proton, and the proton g-factor is:[10]
corresponding to a proton magnetic moment of about μp = 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
The neutron g-factor is:[12]
corresponding to a neutron magnetic moment of about μn = −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
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]
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
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:
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
- ↑ For a number of entertaining exercises on this subject, see Charles Pidgeon (1996). “Larmor frequency versus field strength”, Advanced Tutorials for the Biomedical Sciences: Animations, Simulations, and Calculations Using Mathematica. Wiley-VCH, pp. 165 ff. ISBN 0471186465.
- ↑ 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.
- ↑ Neutron 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. For example, the magnetic moment.
- ↑ 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.
- ↑ 11.0 11.1 See, for example, Brian Martin (2009). “Table 3.5”, Nuclear and Particle Physics: An Introduction, 2nd ed. Wiley, p. 104. ISBN 0470742747. and Steven D. Bass (2008). “Chapter 1: Introduction”, The spin structure of the proton. World Scientific, pp. 1 ff. ISBN 9812709460.
- ↑ Neutron g factor. Fundamental physical constants. NIST. Retrieved on 2011-03-28.
- ↑ For a discussion of the deuteron, see for example S. B. Patel (1991). “§8.2 Ground state of the deuteron”, Nuclear physics: an introduction. New Age International, pp. 276 ff. ISBN 8122401252.
- ↑ Deuteron magnetic moment to nuclear magneton ratio. Fundamental physical constants. NIST. Retrieved on 2011-03-28.
- ↑ For example, see BW Petley (2001). “Chapter 2: The fundamental constants and metrology; §3 The fundamental constants and electrical measurements”, T. J. Quinn, S. Leschiutta, Patrizia Tavella, eds: Metrologia e costanti fondamentali recenti sviluppi: Volume 146 of Proceedings of the International School of Physics "Enrico Fermi". IOS Press, pp. 128 ff. ISBN 1586031678.
- ↑ Robin A. De Graaf (2007). “Chemical shift”, In vivo NMR spectroscopy: principles and techniques, 2nd ed. John Wiley and Sons, p. 18. ISBN 0470026707.