Mathematical joke: Difference between revisions

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29 yr old Orthopaedic Surgeon Grippo from Saint-Paul, spends time with interests including model railways, top property developers in singapore developers in singapore and dolls. Finished a cruise ship experience that included passing by Runic Stones and Church. The magic angle is a precisely defined angle, the value of which is approximately 54.7356°. The magic angle is a root of a second-order Legendre polynomial, $P_{2}(\cos \theta )=0\,$ , 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 magic angle spinning solid-state NMR spectroscopy. In magnetic resonance imaging, structures with ordered collagen, such as tendons and ligaments, oriented at the magic angle may appear hyperintense in some sequences, this is called the magic angle artifact or effect.

Mathematical definition

The magic angle θ_m is

\theta _{m}={\rm {{arccos}{\frac {1}{\sqrt {3}}}={\rm {{arctan}{\sqrt {2}}\approx 0.95532rad\approx 54.7^{\circ }}}}}

,

where arccos and arctan are the inverse cosine and tangent functions respectively Physiotherapist Rave from Cobden, has hobbies and interests which includes skateboarding, commercial property for sale developers in singapore and coin collecting. May be a travel freak and in recent years made a journey to Wet Tropics of Queensland..

θ_m is the angle between the space diagonal of a cube and any of its three connecting edges, see image.

Magic angle and nuclear magnetic resonance

In nuclear magnetic resonance (NMR) spectroscopy, three prominent nuclear magnetic interactions, dipolar coupling, chemical shift anisotropy (CSA), and first-order quadrupolar coupling, depend on the orientation of the interaction tensor with the external magnetic field.

By spinning the sample around a given axis, their average angular dependence becomes:

$\left\langle 3{{\cos }^{2}}\theta -1\right\rangle =\left(3{{\cos }^{2}}{{\theta }_{r}}-1\right)\left(3{{\cos }^{2}}\beta -1\right)$ ,

where $\theta$ is the angle between the principal axis of the interaction and the magnetic field, $\theta _{r}$ is the angle of the axis of rotation relative to the magnetic field and $\beta$ is the (arbitrary) angle between the axis of rotation and principal axis of the interaction.

For dipolar couplings, the principal axis corresponds to the internuclear vector between the coupled spins; for the CSA, it corresponds to the direction with the largest deshielding; for the quadrupolar coupling, it corresponds to the z-axis of the electric-field gradient tensor.

The angle $\beta$ cannot be manipulated as it depends on the orientation of the interaction relative to the molecular frame and on the orientation of the molecule relative to the external field. The angle $\theta _{r}$ , however, can be decided by the experimenter. If one sets $\theta _{r}=\theta _{m}\approx 54.7^{\circ }$ , then the average angular dependence goes to zero. Magic angle spinning is a technique in solid-state NMR spectroscopy which employs this principle to remove or reduce the influence of anisotropic interactions, thereby increasing spectral resolution.

For a time-independent interaction, i.e. heteronuclear dipolar couplings, CSA and first-order quadrupolar couplings, the anisotropic component is greatly reduced and almost suppressed in the limit of fast spinning, i.e. when the spinning frequency is greater than the breadth of the interaction.

The averaging is only close to zero in a first-order perturbation theory treatment; higher order terms cause allowed frequencies at multiples of the spinning frequency to appear, creating spinning side-bands in the spectra.

Time-dependent interactions, such as homonuclear dipolar couplings, are more difficult to average to their isotropic values by magic angle spinning; a network of strongly coupled spins will produce a mixing of spin states during the course of the sample rotation, interfering with the averaging process.

Application to medical imaging: The magic angle artifact

The magic angle artifact refers to the increased signal on sequences with short echo time (TE) (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).^[1] This artifact occurs when the angle such fibers make with the magnetic field is equal to $\theta _{m}$ .

Example: This artifact comes into play when evaluating the rotator cuff tendons of the shoulder. The magic angle effect can create the appearance of supraspinatus tendinitis.

References

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[1]

@@ Line 1: / Line 1: @@
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+{{About|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 54.7356°. 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 [[magic angle spinning]] [[solid-state NMR]] spectroscopy. In [[magnetic resonance imaging]], structures with ordered [[collagen]], such as [[tendon]]s and [[ligament]]s, oriented at the magic angle may appear hyperintense in some sequences, this is called the magic angle artifact or effect.
+==Mathematical definition==
+[[File: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 0.95532 rad \approx 54.7^\circ</math>,
+where arccos and arctan are the inverse cosine and tangent functions respectively {{OEIS2C|A195696}}.
+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 nuclear magnetic resonance==
+In [[nuclear magnetic resonance]] (NMR) spectroscopy, three prominent nuclear magnetic interactions, [[dipolar coupling]], [[chemical shift anisotropy]] (CSA), and [[Solid-state_nuclear_magnetic_resonance#Quadrupolar_interaction|first-order quadrupolar coupling]], depend on the orientation of the interaction tensor with the external magnetic field.
+By spinning the sample around a given axis, their average angular dependence becomes:
+<math>\left\langle 3{{\cos }^{2}}\theta -1 \right\rangle =\left( 3{{\cos }^{2}}{{\theta }_{r}}-1 \right)\left( 3{{\cos }^{2}}\beta -1 \right)</math>,
+where <math>\theta</math> is the angle between the principal axis of the interaction and the magnetic field, <math>\theta_{r}</math> is the angle of the axis of rotation relative to the magnetic field and <math>\beta</math> is the (arbitrary) angle between the axis of rotation and principal axis of the interaction.
+For dipolar couplings, the principal axis corresponds to the internuclear vector between the coupled spins; for the CSA, it corresponds to the direction with the largest deshielding; for the quadrupolar coupling, it corresponds to the z-axis of the electric-field gradient tensor.
+The angle <math>\beta</math> cannot be manipulated as it depends on the orientation of the interaction relative to the molecular frame and on the orientation of the molecule relative to the external field. The angle <math>\theta_r</math>, however, can be decided by the experimenter. If one sets <math>\theta_r = \theta_m \approx 54.7^\circ</math>, then the average angular dependence goes to zero. [[Magic angle spinning]] is a technique in [[solid-state NMR]] spectroscopy which employs this principle to remove or reduce the influence of anisotropic interactions, thereby increasing spectral resolution.
+For a time-independent interaction, i.e. heteronuclear dipolar couplings, CSA and first-order quadrupolar couplings, the anisotropic component is greatly reduced and almost suppressed in the limit of fast spinning, i.e. when the spinning frequency is greater than the breadth of the interaction.
+The averaging is only close to zero in a first-order [[perturbation theory]] treatment; higher order terms cause allowed frequencies at multiples of the spinning frequency to appear, creating spinning side-bands in the spectra.
+Time-dependent interactions, such as homonuclear dipolar couplings, are more difficult to average to their isotropic values by magic angle spinning; a network of strongly coupled spins will produce a mixing of spin states during the course of the sample rotation, interfering with the averaging process.
+==Application to medical imaging: The magic angle artifact==
+[[File:Magic angle artifact.jpg|thumb|Magic angle effect seen on MRI of the shoulder]]
+The magic angle artifact refers to the increased signal on sequences with short echo time (TE) (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>.
+Example: This artifact comes into play when evaluating the rotator cuff tendons of the shoulder. The magic angle effect can create the appearance of supraspinatus tendinitis.
+==References==
+<references/>
+[[Category:Spectroscopy]]
+[[Category:Nuclear magnetic resonance]]
+[[Category:Magnetic resonance imaging]]

Mathematical joke: Difference between revisions

Revision as of 21:28, 5 January 2014

Contents

Mathematical definition

Magic angle and nuclear magnetic resonance

Application to medical imaging: The magic angle artifact

References

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Mathematical joke: Difference between revisions

Revision as of 21:28, 5 January 2014

Mathematical definition

Magic angle and nuclear magnetic resonance

Application to medical imaging: The magic angle artifact

References

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