Magic angle
2289050
223291678
2008-07-03T12:41:26Z
Frap
612852
Changed into template
{{this2|the magic angle as defined in the field of nuclear magnetic resonance spectroscopy|the magic angle as defined in the field of electron energy-loss spectroscopy|magic angle (EELS)}}
The '''magic angle''' is a precisely defined angle, the value of which is approximately 54.7°. The magic angle is a root of a second-order [[Legendre polynomial]], <math>P_2(\cos\theta)=0 \,</math>, and so any interaction which depends on this second-order Legendre polynomial vanishes at the magic angle. This property makes the magic angle of particular importance in [[solid-state NMR]] spectroscopy.
==Mathematical definition==
[[Image:Magic angle.png|thumb|Magic angle|250px|right|Magic angle]]
The magic angle θ<sub>m</sub> is
: <math> \theta_m = \rm{arccos}\frac{1}{\sqrt{3}} = \rm{arctan}\sqrt{2} \approx 54.7^\circ</math>,
where arccos and arctan are the inverse cosine and tangent functions respectively.
θ<sub>m</sub> is the angle between the space diagonal of a cube and any of its three connecting edges, see image.
==Magic angle and dipolar coupling==
In [[nuclear magnetic resonance]] (NMR) spectroscopy, the [[dipolar coupling]] ''D'' in a strong magnetic field depends on the orientation of the internuclear vector with the external magnetic field by
: <math>D(\theta) \propto 3\cos^2\theta - 1</math>
Hence, two nuclei with an internuclear vector at an angle of θ<sub>m</sub> to a strong external magnetic field, have zero dipolar coupling, D(θ<sub>m</sub>)=0. [[Magic angle spinning]] is a technique in [[solid-state NMR]] spectroscopy which employs this principle to remove or reduce dipolar couplings, thereby increasing spectral resolution.
==Application to medical imaging: The magic angle artifact==
The magic angle artifact refers to the increased signal on sequences with short echo time (e.g., T1 or PD Spin Echo sequences ) in MR images seen in tissues with well-ordered collagen fibers in one direction (e.g., tendon or articular hyaline cartilage)<ref>{{cite journal |author=Bydder M, Rahal A, Fullerton G, Bydder G |title=The magic angle effect: a source of artifact, determinant of image contrast, and technique for imaging |journal=Journal of magnetic resonance imaging|volume=25 |issue=2 |pages=290–300 |year=2007 |pmid=17260400 |doi=10.1002/jmri.20850}}</ref>. This artifact occurs when the angle such fibers make with the magnetic field is equal to <math> \theta_m</math>.
==References==
<references/>
[[Category:Materials science]]
[[Category:Spectroscopy]]
[[Category:Radiology]]
[[Category:Nuclear magnetic resonance]]
[[Category:Magnetic resonance imaging]]