Gyromagnetic ratio
Template:Short description Template:Use dmy dates In physics, the gyromagnetic ratio (also sometimes known as the magnetogyric ratio<ref>Template:GreenBook3rd</ref> in other disciplines) of a particle or system is the ratio of its magnetic moment to its angular momentum, and it is often denoted by the symbol [[Gamma|Template:Mvar]], gamma. Its SI unit is the radian per second per tesla (rad⋅s−1⋅T−1) or, equivalently, the coulomb per kilogram (C⋅kg−1).
The [[g-factor (physics)|Template:Mvar-factor]] of a particle is related to its gyromagnetic ratio by a constant multiplier that is related to the system, and is dimensionless.
For a classical rotating bodyEdit
Consider a nonconductive charged body rotating about an axis of symmetry. According to the laws of classical physics, it has both a magnetic dipole moment due to the movement of charge and an angular momentum due to the movement of mass arising from its rotation. It can be shown that as long as its charge and mass densities and currents are distributed identically and rotationally symmetric, its gyromagnetic ratio is
- <math> \gamma = \frac{q}{2m},</math>
where <math>q</math> is its charge, and <math>m</math> is its mass.
The derivation of this relation is as follows. It suffices to demonstrate this for an infinitesimally narrow circular ring within the body, as the general result then follows from an integration. Suppose the ring has radius Template:Mvar, area Template:Math, mass Template:Mvar, charge Template:Mvar, and angular momentum Template:Math. Then the magnitude of the magnetic dipole moment is
- <math> \mu = I A = \frac{q v}{2 \pi r} \, \pi r^2 = \frac{q}{2m} \, m v r = \frac{q}{2m} L.</math>
For an isolated electronEdit
An isolated electron has an angular momentum and a magnetic moment resulting from its spin. While an electron's spin is sometimes visualized as a literal rotation about an axis, it cannot be attributed to mass distributed identically to the charge. The above classical relation does not hold, giving the wrong result by the absolute value of the electron's Template:Mvar-factor, which is denoted Template:Math: <math display="block">
\gamma_\text{e} = \frac{-e}{2 m_\text{e}} \, |g_\mathrm{e}| = \frac{g_\text{e} \mu_\text{B}}{\hbar},
</math> where Template:Math is the Bohr magneton.
The gyromagnetic ratio due to electron spin is twice that due to the orbiting of an electron.
In the framework of relativistic quantum mechanics, <math display="block">
g_\text{e} = -2 \left(1 + \frac{\alpha}{2\pi} + \cdots\right),
</math> where <math>\alpha</math> is the fine-structure constant. Here the small corrections to the relativistic result Template:Math come from the quantum field theory calculations of the anomalous magnetic dipole moment. The electron Template:Mvar-factor is known to twelve decimal places by measuring the electron magnetic moment in a one-electron cyclotron:<ref>Template:Cite journal</ref> <math display="block">
g_\text{e} = -2.002\,319\,304\,361\,18(27).
</math>
The electron gyromagnetic ratio is<ref name="NIST: Electron gyromagnetic ratio"> {{#invoke:citation/CS1|citation |CitationClass=web }} Note that NIST puts a positive sign on the quantity; however, to be consistent with the formulas in this article, a negative sign is put on Template:Mvar here. Indeed, many references say that Template:Nowrap for an electron; for example, Template:Cite bookTemplate:Full citation needed Also note that the units of radians are added for clarity.</ref><ref name="NIST: Electron gyromagnetic ratio value"> {{#invoke:citation/CS1|citation |CitationClass=web }} </ref><ref name="NIST: electron gyromagnetic ratio over 2 pi"> {{#invoke:citation/CS1|citation |CitationClass=web }} </ref> <math display="block">
\gamma_\text{e} = -1.760\,859\,630\,23(53) \times 10^{11}~\text{rad} \cdot \text{s}^{-1} \cdot \text{T}^{-1},
</math> <math display="block">
\frac{\gamma_\text{e}}{2\pi} = -28\,024.951\,4242(85)~ \text{MHz} \cdot \text{T}^{-1}.
</math>
The electron Template:Mvar-factor and Template:Mvar are in excellent agreement with theory; see Precision tests of QED for details.<ref name=Note-1> Template:Cite conference</ref>
Gyromagnetic factor not as a consequence of relativityEdit
Since a gyromagnetic factor equal to 2 follows from Dirac's equation, it is a frequent misconception to think that a Template:Mvar-factor 2 is a consequence of relativity; it is not. The factor 2 can be obtained from the linearization of both the Schrödinger equation (known as Lévy-Leblond equation) and the relativistic Klein–Gordon equation (which leads to Dirac's). In both cases a 4-spinor is obtained and for both linearizations the Template:Mvar-factor is found to be equal to 2. Therefore, the factor 2 is a consequence of the minimal coupling and of the fact of having the same order of derivatives for space and time.<ref> Template:Cite book </ref>
Physical spin-Template:Sfrac particles which cannot be described by the linear gauged Dirac equation satisfy the gauged Klein–Gordon equation extended by the Template:Math term according to,<ref name="fourvectorspinor" />
- <math>
\left[\left(\partial^\mu + i\, e\, A^\mu\right) \left(\partial_\mu + i\, e\, A_\mu \right) +
g \, \frac{e}{4} \, \sigma^{\mu\nu} \, F_{\mu\nu} + m^2\right] \psi = 0, \quad g \ne 2.
</math>
Here, Template:Math and Template:Math stand for the Lorentz group generators in the Dirac space, and the electromagnetic tensor respectively, while Template:Math is the electromagnetic four-potential. An example for such a particle<ref name="fourvectorspinor"> Template:Cite journal </ref> is the spin Template:Sfrac companion to spin Template:Sfrac in the Template:Math representation space of the Lorentz group. This particle has been shown to be characterized by Template:Nowrap and consequently to behave as a truly quadratic fermion.
For a nucleusEdit
Protons, neutrons, and many nuclei carry nuclear spin, which gives rise to a gyromagnetic ratio as above. The ratio is conventionally written in terms of the proton mass and charge, even for neutrons and for other nuclei, for the sake of simplicity and consistency. The formula is:
- <math> \gamma_\text{n} = \frac{e}{2m_\text{p}} \, g_\text{n} = g_\text{n}\, \frac{\mu_\text{N}}{\hbar},</math>
where <math>\mu_\text{N}</math> is the nuclear magneton, and <math>g_\text{n}</math> is the [[g-factor (physics)|Template:Mvar-factor]] of the nucleon or nucleus in question. The ratio <math>\frac{\gamma_n}{2 \pi \, g_\text{n}} = \mu_\text{N}/h = 7.622593285(47)</math> MHz/T.<ref> {{#invoke:citation/CS1|citation |CitationClass=web }} (citing CODATA-recommended values) </ref>
The gyromagnetic ratio of a nucleus plays a role in nuclear magnetic resonance (NMR) and magnetic resonance imaging (MRI). These procedures rely on the fact that bulk magnetization due to nuclear spins precess in a magnetic field at a rate called the Larmor frequency, which is simply the product of the gyromagnetic ratio with the magnetic field strength. With this phenomenon, the sign of Template:Mvar determines the sense (clockwise vs counterclockwise) of precession.
Most common nuclei such as 1H and 13C have positive gyromagnetic ratios.<ref name="M H Levitt 2008"> Template:Cite book </ref><ref name="Arthur G Palmer 2007"> Template:Cite book </ref> Approximate values for some common nuclei are given in the table below.<ref> Template:Cite book</ref><ref> Template:Cite book </ref>
| Nucleus | <math>\gamma_n</math> (106 rad⋅s−1⋅T−1) | <math>\gamma_n/(2\pi)</math> (MHz⋅T−1) | ||
|---|---|---|---|---|
| 1H | Template:Val<ref>
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Template:Val<ref>
{{#invoke:citation/CS1|citation |
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}} </ref> |
| 1H (in H2O) | Template:Val<ref>{{#invoke:citation/CS1|citation | CitationClass=web
}}</ref> |
Template:Val<ref>{{#invoke:citation/CS1|citation | CitationClass=web
}}</ref> |
| 2H | 41.065 | 6.536 | ||
| 3H | 285.3508 | 45.415<ref>
{{#invoke:citation/CS1|citation |
CitationClass=web
}} </ref> | |
| 3He | Template:Val<ref>{{#invoke:citation/CS1|citation | CitationClass=web
}}</ref> |
Template:Val<ref>{{#invoke:citation/CS1|citation | CitationClass=web
}}</ref> |
| 7Li | 103.962 | 16.546 | ||
| 13C | 67.2828 | 10.7084 | ||
| 14N | 19.331 | 3.077 | ||
| 15N | −27.116 | −4.316 | ||
| 17O | −36.264 | −5.772 | ||
| 19F | 251.815 | 40.078 | ||
| 23Na | 70.761 | 11.262 | ||
| 27Al | 69.763 | 11.103 | ||
| 29Si | −53.190 | −8.465 | ||
| 31P | 108.291 | 17.235 | ||
| 57Fe | 8.681 | 1.382 | ||
| 63Cu | 71.118 | 11.319 | ||
| 67Zn | 16.767 | 2.669 | ||
| 129Xe | Template:Val | Template:Val<ref>Template:Cite journal</ref> |
A full list can be found in the external link section below.
Larmor precessionEdit
Template:Main article Any free system with a constant gyromagnetic ratio, such as a rigid system of charges, a nucleus, or an electron, when placed in an external magnetic field Template:Math (measured in teslas) that is not aligned with its magnetic moment, will precess at a frequency Template:Mvar (measured in hertz) proportional to the external field:
- <math>f = \frac{\gamma}{2\pi} B.</math>
For this reason, values of Template:Sfrac, in units of hertz per tesla (Hz/T), are often quoted instead of Template:Mvar.
Heuristic derivationEdit
The derivation of this ratio is as follows: First we must prove the torque resulting from subjecting a magnetic moment <math>\mathbf{m}</math> to a magnetic field <math>\mathbf{B}</math> is <math>\boldsymbol{\Tau} = \mathbf{m} \times \mathbf{B}.</math> The identity of the functional form of the stationary electric and magnetic fields has led to defining the magnitude of the magnetic dipole moment equally well as <math>m=I\pi r^2</math>, or in the following way, imitating the moment Template:Math of an electric dipole: The magnetic dipole can be represented by a needle of a compass with fictitious magnetic charges <math>\pm q_\text{m}</math> on the two poles and vector distance between the poles <math>\mathbf{d}</math> under the influence of the magnetic field of earth <math>\mathbf{B}.</math> By classical mechanics the torque on this needle is <math>\boldsymbol{\Tau} = q_\text{m}(\mathbf{d} \times \mathbf{B}).</math> But as previously stated <math>q_\text{m} \mathbf{d} = I\pi r^2 \hat{\mathbf{d}} = \mathbf{m},</math> so the desired formula comes up. <math>\hat{\mathbf{d}}</math> is the unit distance vector.
The spinning electron model here is analogous to a gyroscope. For any rotating body the rate of change of the angular momentum <math>\mathbf{J}</math> equals the applied torque <math>\mathbf{T}</math>:
- <math>\frac{d\mathbf{J}}{dt} = \mathbf{T}.</math>
Note as an example the precession of a gyroscope. The earth's gravitational attraction applies a force or torque to the gyroscope in the vertical direction, and the angular momentum vector along the axis of the gyroscope rotates slowly about a vertical line through the pivot. In place of a gyroscope, imagine a sphere spinning around the axis with its center on the pivot of the gyroscope, and along the axis of the gyroscope two oppositely directed vectors both originated in the center of the sphere, upwards <math>\mathbf{J}</math> and downwards <math>\mathbf{m}.</math> Replace the gravity with a magnetic flux density <math>\mathbf{B}.</math>
<math>\frac{d\mathbf{J}}{dt}</math> represents the linear velocity of the pike of the arrow <math>\mathbf{J}</math> along a circle whose radius is <math>J\sin{\phi},</math> where <math>\phi</math> is the angle between <math>\mathbf{J}</math> and the vertical. Hence the angular velocity of the rotation of the spin is
- <math>\omega = 2\pi \,f = \frac{1}{J \sin{\phi}} \left|\frac{d\mathbf{J}}{dt}\right|
= \frac{|\mathbf{T}|}{J \sin{\phi}} = \frac{|\mathbf{m} \times \mathbf{B}|}{J \sin{\phi}}
= \frac{m\,B\sin{\phi}}{J \sin{\phi}} = \frac{m\, B}{J} = \gamma\, B.</math>
Consequently, <math>f = \frac{\gamma}{2\pi}\,B,\quad \text{q.e.d.}</math>
This relationship also explains an apparent contradiction between the two equivalent terms, gyromagnetic ratio versus magnetogyric ratio: whereas it is a ratio of a magnetic property (i.e. dipole moment) to a gyric (rotational, from Template:Langx, "turn") property (i.e. angular momentum), it is also a ratio between the angular precession frequency (another gyric property) Template:Math and the magnetic field.
The angular precession frequency has an important physical meaning: It is the angular cyclotron frequency, the resonance frequency of an ionized plasma being under the influence of a static finite magnetic field, when we superimpose a high frequency electromagnetic field.
See alsoEdit
- Charge-to-mass ratio
- Chemical shift
- [[Landé g-factor|Landé Template:Mvar-factor]]
- Larmor equation
- Proton gyromagnetic ratio
ReferencesEdit
External linksEdit
- Konstantin's gyromagnetic ratio table A full list of all known gyromagnetic ratios.