메뉴 건너뛰기




Volumn 9, Issue 3, 1997, Pages 149-171

Rotations in NMR: Part I. Euler - Rodrigues parameters and quaternions

Author keywords

Cayley Klein parameters; Composite rotations; Euler Rodrigues parameters; Euler angles; Kinematic relations; Quaternions; SO(3); Spinors; SU(2)

Indexed keywords


EID: 0031286136     PISSN: 10437347     EISSN: None     Source Type: Journal    
DOI: 10.1002/(SICI)1099-0534(1997)9:3<149::AID-CMR3>3.0.CO;2-Z     Document Type: Article
Times cited : (50)

References (62)
  • 3
    • 0000837280 scopus 로고
    • Quaternions: A practical tool for the evaluation of composite rotations
    • B. Blumich and H. Spiess, "Quaternions: A Practical Tool for the Evaluation of Composite Rotations," J. Magn. Reson., 1985, 61, 356-362.
    • (1985) J. Magn. Reson. , vol.61 , pp. 356-362
    • Blumich, B.1    Spiess, H.2
  • 4
  • 5
    • 0000745033 scopus 로고
    • Analytical expressions and line-shape simulations for Levitt-Suter-Ernst composite-pulse quadrupolar echo sequences
    • T. M. Barbara, "Analytical Expressions and Line-shape Simulations for Levitt-Suter-Ernst Composite-Pulse Quadrupolar Echo Sequences," J. Magn. Reson., 1986, 67, 491-500.
    • (1986) J. Magn. Reson. , vol.67 , pp. 491-500
    • Barbara, T.M.1
  • 6
    • 0000603766 scopus 로고
    • Frequency response of soft pulses
    • M. Shinnar and J. S. Leigh, "Frequency Response of Soft Pulses," J. Magn. Reson., 1987, 75, 502-505.
    • (1987) J. Magn. Reson. , vol.75 , pp. 502-505
    • Shinnar, M.1    Leigh, J.S.2
  • 7
    • 33748476849 scopus 로고
    • Suppression of cross-relaxation effects in TOCSY spectra via a modified DIPSI-2 mixing sequence
    • J. Cavanagh and M. Rance, "Suppression of Cross-Relaxation Effects in TOCSY Spectra via a Modified DIPSI-2 Mixing Sequence," J. Magn. Reson., 1992, 96, 670-678.
    • (1992) J. Magn. Reson. , vol.96 , pp. 670-678
    • Cavanagh, J.1    Rance, M.2
  • 8
    • 0002593105 scopus 로고
    • Optimization of shaped selective pulses for NMR using a quaternion description of their overall propagators
    • L. Emsley and G. Bodenhausen, "Optimization of Shaped Selective Pulses for NMR Using a Quaternion Description of Their Overall Propagators," J. Magn. Reson., 1992, 97, 135-148.
    • (1992) J. Magn. Reson. , vol.97 , pp. 135-148
    • Emsley, L.1    Bodenhausen, G.2
  • 9
    • 0000525694 scopus 로고
    • Simulation of multipulse tune-up sequences
    • T. M. Barbara, "Simulation of Multipulse Tune-Up Sequences," J. Magn. Reson. A, 1994, 106, 188-194.
    • (1994) J. Magn. Reson. A , vol.106 , pp. 188-194
    • Barbara, T.M.1
  • 11
    • 0642270490 scopus 로고
    • Exponen tial operator approach to the general solution to off-resonance RF field interaction
    • J. Zhou, H. Gao, and B.C. Sanctuary, "Exponen tial Operator Approach to the General Solution to Off-Resonance RF Field Interaction," J. Magn. Reson. A, 1993, 707, 119-121.
    • (1993) J. Magn. Reson. A , vol.707 , pp. 119-121
    • Zhou, J.1    Gao, H.2    Sanctuary, B.C.3
  • 12
    • 0000142790 scopus 로고
    • Rotation-operator approach and spin dynamics in a time-varying magnetic field
    • J. Zhou and C. Ye, "Rotation-Operator Approach and Spin Dynamics in a Time-Varying Magnetic Field," Phys. Rev. A, 1994, 50, 1903-1904.
    • (1994) Phys. Rev. A , vol.50 , pp. 1903-1904
    • Zhou, J.1    Ye, C.2
  • 13
    • 0001346250 scopus 로고
    • Rotation operator approach to spin dynamics and the euler geometric equations
    • J. Zhou, C. Ye, and B.C. Sanctuary, "Rotation Operator Approach to Spin Dynamics and the Euler Geometric Equations," J. Chem. Phys., 1994, 101, 6424-6429.
    • (1994) J. Chem. Phys. , vol.101 , pp. 6424-6429
    • Zhou, J.1    Ye, C.2    Sanctuary, B.C.3
  • 14
    • 0000730543 scopus 로고
    • Explicit demonstration of spinor character for a spin-1/2 nucleus via NMR interferometry
    • M. E. Stoll, A. J. Vega, and R. W. Vaughan, "Explicit Demonstration of Spinor Character for a Spin-1/2 Nucleus via NMR Interferometry," Phys. Rev. A, 1977, 76, 1521-1524.
    • (1977) Phys. Rev. A , vol.76 , pp. 1521-1524
    • Stoll, M.E.1    Vega, A.J.2    Vaughan, R.W.3
  • 15
    • 4243613369 scopus 로고
    • Indirect phase detection of NMR spinor-transitions
    • D. Suter, A. Pines, and M. Mehring, "Indirect Phase Detection of NMR Spinor-Transitions," Phys. Rev. Lett., 1986, 57, 242-244.
    • (1986) Phys. Rev. Lett. , vol.57 , pp. 242-244
    • Suter, D.1    Pines, A.2    Mehring, M.3
  • 16
    • 4243491515 scopus 로고
    • Application of spinors to solving the bloch equations
    • G. Town and D. Rosenfeld, "Application of Spinors to Solving the Bloch Equations," Phys. Rev. A, 1989, 40, 3429-3430.
    • (1989) Phys. Rev. A , vol.40 , pp. 3429-3430
    • Town, G.1    Rosenfeld, D.2
  • 17
    • 0010923380 scopus 로고
    • The application of spinors to solving the bloch equations
    • C. Barratt, E. D. Fackerell, and D. Rosenfeld, "The Application of Spinors to Solving the Bloch Equations," J. Magn. Reson., 1989, 85, 35-41.
    • (1989) J. Magn. Reson. , vol.85 , pp. 35-41
    • Barratt, C.1    Fackerell, E.D.2    Rosenfeld, D.3
  • 18
    • 0000488661 scopus 로고
    • Matrix treatment of nuclear induction
    • E. T. Jaynes, "Matrix Treatment of Nuclear Induction," Phys. Rev., 1955, 98, 1099-1105.
    • (1955) Phys. Rev. , vol.98 , pp. 1099-1105
    • Jaynes, E.T.1
  • 19
    • 11744315676 scopus 로고
    • Vectors, spinors, and complex numbers in classical and quantum physics
    • D. Hestenes, "Vectors, Spinors, and Complex Numbers in Classical and Quantum Physics," Am. J. Phys., 1971, 39, 1013-1027.
    • (1971) Am. J. Phys. , vol.39 , pp. 1013-1027
    • Hestenes, D.1
  • 20
    • 21144473471 scopus 로고
    • An introduction to geometric algebra with an application in rigid body mechanics
    • T. G. Vold, "An Introduction to Geometric Algebra with an Application in Rigid Body Mechanics," Am. J. Phys., 1993, 61, 491-504.
    • (1993) Am. J. Phys. , vol.61 , pp. 491-504
    • Vold, T.G.1
  • 21
    • 0000613202 scopus 로고
    • On quaternions; or on a new system of imaginaries in algebra
    • W. R. Hamilton, "On Quaternions; or on a New System of Imaginaries in Algebra," Phil. Mag., 3rd Ser., 1844, 25, 489-495.
    • (1844) Phil. Mag., 3rd Ser. , vol.25 , pp. 489-495
    • Hamilton, W.R.1
  • 22
    • 0000741026 scopus 로고
    • Des lois géométriques qui régissent les déplacements d'un systéme solide dans l'espace et de la variation des coordonnées provenant de ses déplacements considérés indépendamment des causes qui peuvent les produire
    • O. Rodrigues, "Des Lois Géométriques qui Régissent les Déplacements d'un Systéme Solide dans L'espace et de la Variation des Coordonnées Provenant de ses Déplacements Considérés Indépendamment des Causes qui Peuvent les Produire," J. Math. Pures Appl., 1840, 5, 380-440.
    • (1840) J. Math. Pures Appl. , vol.5 , pp. 380-440
    • Rodrigues, O.1
  • 23
    • 0010928885 scopus 로고
    • Quaternions as 4-vectors
    • M. J. Walker, "Quaternions as 4-Vectors," Am. J. Phys., 1956, 24, 515-522.
    • (1956) Am. J. Phys. , vol.24 , pp. 515-522
    • Walker, M.J.1
  • 24
    • 84975553202 scopus 로고
    • Closed-form solution of absolute orientation using unit quaternions
    • B.K.P. Horn, "Closed-Form Solution of Absolute Orientation Using Unit Quaternions," J. Opt. Soc. Am. A, 1987, 4, 629-642.
    • (1987) J. Opt. Soc. Am. A , vol.4 , pp. 629-642
    • Horn, B.K.P.1
  • 25
    • 0000369182 scopus 로고
    • On certain results relating to quaternions
    • A. Cayley, "On Certain Results Relating to Quaternions," Phil. Mag., 3rd Ser., 1845, 26, 141-145.
    • (1845) Phil. Mag., 3rd Ser. , vol.26 , pp. 141-145
    • Cayley, A.1
  • 27
    • 0003194420 scopus 로고
    • Angular momentum in quantum physics: Theory and application
    • Addison-Wesley, Reading, MA, Ch. 2, 4
    • L. C. Biedenharn and J. D. Louck, "Angular Momentum in Quantum Physics: Theory and Application," In Encyclopedia of Mathematics and its Applications, Vol. 8, Addison-Wesley, Reading, MA, 1981, Ch. 2, 4.
    • (1981) Encyclopedia of Mathematics and Its Applications , vol.8
    • Biedenharn, L.C.1    Louck, J.D.2
  • 29
    • 0010928886 scopus 로고
    • Representation of rotations in three-space by complex 2 × 2 matrices
    • M. J. Walker, "Representation of Rotations in Three-Space by Complex 2 × 2 Matrices," Am. J. Phys., 1956, 24, 205-208.
    • (1956) Am. J. Phys. , vol.24 , pp. 205-208
    • Walker, M.J.1
  • 30
    • 0000411371 scopus 로고
    • A slice-selective adiabatic refocusing pulse made up of two adiabatic inversion pulses. Suppression of unwanted echoes
    • J. F. Shen and J. K. Saunders, "A Slice-Selective Adiabatic Refocusing Pulse Made Up of Two Adiabatic Inversion Pulses. Suppression of Unwanted Echoes," J. Magn. Reson., 1992, 96, 381-386.
    • (1992) J. Magn. Reson. , vol.96 , pp. 381-386
    • Shen, J.F.1    Saunders, J.K.2
  • 31
    • 0038060311 scopus 로고
    • On the rotation operators in quantum mechanics
    • M. Bouten, "On the Rotation Operators in Quantum Mechanics," Physica, 1969, 42, 572-580.
    • (1969) Physica , vol.42 , pp. 572-580
    • Bouten, M.1
  • 34
    • 36149018966 scopus 로고
    • Use of rotating coordinates in magnetic resonance problems
    • I. I. Rabi, N. F. Ramsey, and J. Schwinger, "Use of Rotating Coordinates in Magnetic Resonance Problems," Rev. Mod. Phys., 1954, 26, 167-171.
    • (1954) Rev. Mod. Phys. , vol.26 , pp. 167-171
    • Rabi, I.I.1    Ramsey, N.F.2    Schwinger, J.3
  • 35
    • 0001385876 scopus 로고
    • Vector representation of rigid body rotation
    • C. Grubin, "Vector Representation of Rigid Body Rotation," Am. J. Phys., 1962, 30, 416-417.
    • (1962) Am. J. Phys. , vol.30 , pp. 416-417
    • Grubin, C.1
  • 38
    • 0021957281 scopus 로고
    • Theoretical description of depth pulse sequences, on and off resonance, including improvements and extensions thereof
    • M. R. Bendall and D. T. Pegg, "Theoretical Description of Depth Pulse Sequences, On and Off Resonance, Including Improvements and Extensions Thereof," Magn. Reson. Med., 1985, 2, 91-113.
    • (1985) Magn. Reson. Med. , vol.2 , pp. 91-113
    • Bendall, M.R.1    Pegg, D.T.2
  • 40
    • 48749144559 scopus 로고
    • Evaluation of a new broadband decoupling sequence: WALTZ-16
    • A. J. Shaka, J. Keeler, and R. Freeman, "Evaluation of a New Broadband Decoupling Sequence: WALTZ-16," J. Magn. Reson., 1983, 55, 313-340.
    • (1983) J. Magn. Reson. , vol.55 , pp. 313-340
    • Shaka, A.J.1    Keeler, J.2    Freeman, R.3
  • 41
    • 0001346250 scopus 로고
    • Comment on 'rotation operator approach to spin dynamics and the euler geometric equations'
    • D. J. Siminovitch, "Comment on 'Rotation Operator Approach to Spin Dynamics and the Euler Geometric Equations' [J. Chem. Phys. 101, 6424 (1994)]," J. Chem. Phys., 1995, 103, 2766-2768.
    • (1994) J. Chem. Phys. , vol.101 , pp. 6424
    • Siminovitch, D.J.1
  • 42
    • 0010796748 scopus 로고
    • D. J. Siminovitch, "Comment on 'Rotation Operator Approach to Spin Dynamics and the Euler Geometric Equations' [J. Chem. Phys. 101, 6424 (1994)]," J. Chem. Phys., 1995, 103, 2766-2768.
    • (1995) J. Chem. Phys. , vol.103 , pp. 2766-2768
  • 43
    • 36149021337 scopus 로고
    • Nuclear induction in inhomogeneous fields
    • A. L. Bloom, "Nuclear Induction in Inhomogeneous Fields," Phys. Rev., 1955, 98, 1105-1111.
    • (1955) Phys. Rev. , vol.98 , pp. 1105-1111
    • Bloom, A.L.1
  • 44
    • 0002428345 scopus 로고
    • Theory of broadband spin decoupling
    • J. S. Waugh, "Theory of Broadband Spin Decoupling," J Magn. Reson., 1982, 50, 30-49.
    • (1982) J Magn. Reson. , vol.50 , pp. 30-49
    • Waugh, J.S.1
  • 45
    • 0000435342 scopus 로고
    • Exact solutions for selective-excitation pulses
    • J. W. Carlson, "Exact Solutions for Selective-Excitation Pulses," J. Magn. Reson., 1991, 94, 376-386.
    • (1991) J. Magn. Reson. , vol.94 , pp. 376-386
    • Carlson, J.W.1
  • 46
    • 51249186915 scopus 로고
    • On prehistory of spin kinematics
    • A. Turrin, "On Prehistory of Spin Kinematics," Lett. Nuov. Ciment., 1982, 35, 129-131.
    • (1982) Lett. Nuov. Ciment. , vol.35 , pp. 129-131
    • Turrin, A.1
  • 48
    • 0000289963 scopus 로고
    • The utility and limitations of the spinorized bloch equation
    • A. Takahashi, T. Peters, and Y. Takahashi, "The Utility and Limitations of the Spinorized Bloch Equation," J. Magn. Reson., 1992, 98, 147-152.
    • (1992) J. Magn. Reson. , vol.98 , pp. 147-152
    • Takahashi, A.1    Peters, T.2    Takahashi, Y.3
  • 49
    • 0000968918 scopus 로고
    • Composite pulses for ultra-broadband spin inversion
    • A. J. Shaka, "Composite Pulses for Ultra-Broadband Spin Inversion," Chem. Phys. Lett., 1985, 120, 201-205.
    • (1985) Chem. Phys. Lett. , vol.120 , pp. 201-205
    • Shaka, A.J.1
  • 51
    • 0002715617 scopus 로고
    • A classical view of the euler angles and the euler kinematic equations in NMR
    • D. J. Siminovitch, "A Classical View of the Euler Angles and the Euler Kinematic Equations in NMR," J. Magn. Reson. A, 1995, 117, 235-245.
    • (1995) J. Magn. Reson. A , vol.117 , pp. 235-245
    • Siminovitch, D.J.1
  • 52
    • 0010864969 scopus 로고
    • Operator techniques in the phenomenological treatment of magnetic resonance
    • P. L. Corio, "Operator Techniques in the Phenomenological Treatment of Magnetic Resonance," J. Chem. Phys., 1963, 38, 979-988.
    • (1963) J. Chem. Phys. , vol.38 , pp. 979-988
    • Corio, P.L.1
  • 53
    • 33747316988 scopus 로고
    • Lie algebraic solution of bloch equations with time-dependent coefficients
    • S. V. Prants, "Lie Algebraic Solution of Bloch Equations with Time-Dependent Coefficients," Phys. Lett. A, 1990, 144, 225-228.
    • (1990) Phys. Lett. A , vol.144 , pp. 225-228
    • Prants, S.V.1
  • 54
    • 0001727848 scopus 로고
    • On the parameterization of the three-dimensional rotation group
    • J. Stuelpnagel, "On the Parameterization of the Three-Dimensional Rotation Group," SIAM Rev., 1964, 6, 422-430.
    • (1964) SIAM Rev. , vol.6 , pp. 422-430
    • Stuelpnagel, J.1
  • 55
    • 0022863386 scopus 로고
    • Euler parameters and the use of quaternion algebra in the manipulation of finite rotations: A review
    • K. W. Spring, "Euler Parameters and the Use of Quaternion Algebra in the Manipulation of Finite Rotations: A Review," Mech. Mach. Theory, 1986, 27, 365-373.
    • (1986) Mech. Mach. Theory , vol.27 , pp. 365-373
    • Spring, K.W.1
  • 56
    • 0000066126 scopus 로고
    • A guide to rotations in quantum mechanics
    • M. A. Morrison and G. A. Parker, "A Guide to Rotations in Quantum Mechanics," Aust. J. Phys., 1987, 40, 465-497.
    • (1987) Aust. J. Phys. , vol.40 , pp. 465-497
    • Morrison, M.A.1    Parker, G.A.2
  • 57
    • 0015599953 scopus 로고
    • Solution of kinematical differential equations for a rigid body
    • T. R. Kane, "Solution of Kinematical Differential Equations for a Rigid Body," J. Appl. Mech., 1973, 40, 109-113.
    • (1973) J. Appl. Mech. , vol.40 , pp. 109-113
    • Kane, T.R.1
  • 58
    • 0010797217 scopus 로고
    • Notes on quaternions
    • G. Boole, "Notes on Quaternions," Phil. Mag., 3rd Ser., 1848, 33, 278-280.
    • (1848) Phil. Mag., 3rd Ser. , vol.33 , pp. 278-280
    • Boole, G.1
  • 59
    • 0010795420 scopus 로고
    • On the geometrical theory of rotation
    • W. F. Donkin, "On the Geometrical Theory of Rotation," Phil. Mag., 4th Ser., 1851, 1, 187-192.
    • (1851) Phil. Mag., 4th Ser. , vol.1 , pp. 187-192
    • Donkin, W.F.1
  • 60
    • 0000623669 scopus 로고
    • On the representation of orientation space
    • D. J. Evans, "On the Representation of Orientation Space," Molec. Phys., 1977, 34, 317-325.
    • (1977) Molec. Phys. , vol.34 , pp. 317-325
    • Evans, D.J.1
  • 61
    • 0010923382 scopus 로고
    • A modern presentation of quaternions
    • F. D. Murnaghan, "A Modern Presentation of Quaternions," Proc. R. Irish Acad. A, 1945, 50, 104-121.
    • (1945) Proc. R. Irish Acad. A , vol.50 , pp. 104-121
    • Murnaghan, F.D.1
  • 62
    • 0010865545 scopus 로고
    • Generalized gantmacher formulas through functions of matrices
    • L. Y. Bahar, "Generalized Gantmacher Formulas through Functions of Matrices," Am. J. Phys., 1991, 59, 1103-1111.
    • (1991) Am. J. Phys. , vol.59 , pp. 1103-1111
    • Bahar, L.Y.1


* 이 정보는 Elsevier사의 SCOPUS DB에서 KISTI가 분석하여 추출한 것입니다.