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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
512:
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 width of the interaction.
439:
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the fibres must be positioned under an angle of approximately 54.7 angular degrees, also referred to as the magic angle. The magic angle of 54.7 exactly balances the internal-pressure-induced longitudinal stress and the hoop (circumferential) stress.
463:
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
306:
99:
19:
This article is about the magic angle as defined in the field of nuclear magnetic resonance spectroscopy. For the magic angle as defined in the field of electron energy-loss spectroscopy, see
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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
48:, 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
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fibers in one direction (e.g., tendon or articular hyaline cartilage). This artifact occurs when the angle such fibers make with the magnetic field is equal to
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Another representation of the magic angle is half of the opening angle formed when a cube is rotated from its space diagonal axis, which may be represented as
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treatment; higher order terms cause allowed frequencies at multiples of the spinning frequency to appear, creating spinning side-bands in the spectra.
295:
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spectroscopy which employs this principle to remove or reduce the influence of anisotropic interactions, thereby increasing spectral resolution.
434:{\displaystyle \left\langle 3\cos ^{2}\theta -1\right\rangle =\left(3\cos ^{2}\theta _{\mathrm {r} }-1\right)\left(3\cos ^{2}\beta -1\right)\!,}
193:{\displaystyle \theta _{\mathrm {m} }=\arccos {\frac {-1}{3}}=\arctan {\sqrt {2}}\approx 0.955\,32\ {\text{rad}}\approx 54.7^{\circ }\!,}
204:
71:, oriented at the magic angle may appear hyperintense in some sequences; this is called the magic angle artifact or effect.
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30:
is a precisely defined angle, the value of which is approximately 54.7356°. The magic angle is a root of a second-order
672:
259:
610:"The magic angle effect: a source of artifact, determinant of image contrast, and technique for imaging"
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is the angle between the space diagonal of a cube and any of its three connecting edges, see image.
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is the (arbitrary) angle between the axis of rotation and principal axis of the interaction.
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298:, depend on the orientation of the interaction tensor with the external magnetic field.
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By spinning the sample around a given axis, their average angular dependence becomes:
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or proton density spin-echo sequences) are used to image tissues with well-ordered
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is the angle between the principal axis of the interaction and the magnetic field,
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where arccos and arctan are the inverse cosine and tangent functions respectively.
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tendons of the shoulder. The magic angle effect can create the appearance of
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is the angle of the axis of rotation relative to the magnetic field and
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The magic angle artifact refers to the increased signal observed when
266:(i.e., the edge central angle also known as the tetrahedral angle).
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286:(NMR) spectroscopy, three prominent nuclear magnetic interactions,
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and is the angle between two vertices and the exact center of a
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Example: This artifact comes into play when evaluating the
484:, however, can be decided by the experimenter. If one sets
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Application to medical imaging: The magic angle artifact
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The averaging is only close to zero in a first-order
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102:
501:, then the average angular dependence goes to zero.
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192:
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258:. This double magic angle is directly related to
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583:To achieve optimal loading in a straight rubber
278:Magic angle effect seen on MRI of the shoulder
468:-axis of the electric-field gradient tensor.
270:Magic angle and nuclear magnetic resonance
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663:Nuclear magnetic resonance spectroscopy
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604:Bydder, M.; Rahal, A.; Fullerton, G.;
614:Journal of Magnetic Resonance Imaging
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536:with short echo time (TE) (e.g.,
296:first-order quadrupolar coupling
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260:tetrahedral molecular geometry
1:
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7:
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668:Magnetic resonance imaging
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284:nuclear magnetic resonance
59:, structures with ordered
57:magnetic resonance imaging
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292:chemical shift anisotropy
567:supraspinatus tendinitis
75:Mathematical definition
673:Mathematical constants
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16:Concept in mathematics
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503:Magic angle spinning
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50:magic angle spinning
518:perturbation theory
246:2 arctan
32:Legendre polynomial
627:10.1002/jmri.20850
505:is a technique in
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21:magic angle (EELS)
579:Reinforced rubber
573:Reinforced rubber
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55:spectroscopy. In
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620:(2): 290–300.
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534:MRI sequences
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227:arccos −
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563:rotator cuff
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499:≈ 54.7°
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294:(CSA), and
264:tetrahedron
83:Magic angle
28:magic angle
657:Categories
606:Bydder, G.
592:References
471:The angle
63:, such as
524:process.
417:−
414:β
411:
382:−
371:θ
367:
335:−
332:θ
329:
182:∘
174:≈
156:≈
146:
128:−
122:
105:θ
69:ligaments
636:17260400
608:(2007).
547:collagen
342:⟩
312:⟨
254:radians
61:collagen
644:6464946
248:√
241:
229:
209:A195696
207::
65:tendons
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634:
443:where
166:
143:arctan
119:arccos
640:S2CID
159:0.955
46:) = 0
42:(cos
632:PMID
585:hose
205:OEIS
178:54.7
67:and
26:The
622:doi
402:cos
358:cos
320:cos
282:In
244:or
170:rad
94:is
659::
638:.
630:.
618:25
616:.
612:.
569:.
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290:,
163:32
34:,
646:.
624::
555:m
552:θ
542:1
539:T
497:m
494:θ
490:r
487:θ
481:r
478:θ
473:β
466:z
458:β
453:r
450:θ
445:θ
429:,
424:)
420:1
406:2
398:3
394:(
389:)
385:1
376:r
362:2
354:3
350:(
346:=
338:1
324:2
316:3
250:2
238:3
235:/
232:1
219:m
216:θ
188:,
151:2
140:=
135:3
131:1
116:=
110:m
92:m
89:θ
44:θ
40:2
37:P
23:.
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