Wigner rotation matrices for second-rank spherical tensor.
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Wigner rotation matrices for second-rank spherical tensor


A. Wigner active rotation matrix

For static crystal, the orientation of the static magnetic field B0 in the principal-axis system (主軸坐標系) of the EFG tensor (XPAS, YPAS, ZPAS) is described with the Euler angles α, β, and γ (see EasySpin, Wikipedia, Wolfram MathWorld).

Euler angles of the static magnetic field in the laboratory frame

The component V(2,k) of the second-rank spherical tensor V in the obs coordinate frame is:

EFG tensor component V(2,k) Equation 1

The Wigner matrix for active rotation is:

EFG tensor component V(2,k)

Reference

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    Maple procedures for the coupling of angular momenta. IX. Wigner D-functions and rotation matrices,
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  4. Mattias Edén
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    Concepts Magn. Reson. 17A, 117-154 (2003).
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    Euler angles

    The transformation of the coordinate system F into the system F' is parametrized by the triplet of Euler angles {αFF', βFF', γFF'}. The subscripts FF' specify a transformation from system F to system F'.

     
  5. Bernhard Blümich
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    Clarenton, Oxford (2000).
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  8. O. N. Antzutkin
    Sideband manipulation in magic-angle-spinning nuclear magnetic resonance,
    Prog. Nucl. Magn. Reson. Spectrosc. 35, 203-266 (1999).
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    Euler angles

    Euler angles in the rotational transformation from a frame A to a frame B. The unitary transformation of a spherical tensor can be expressed as

    Tensor transformation with Wigner matrix
     
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    The Eulerian angles β and α simply are the polar angles of the z axis in the principal axes system X, Y, Z.

     
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B. Wigner passive rotation matrix

For static crystal, the orientation of the principal-axis system (主軸坐標系) of the EFG tensor (XPAS, YPAS, ZPAS) in the static magnetic field B0 (xobs, yobs, zobs) is described with the Euler angles α, β, and γ.

Euler angles of the static magnetic field in the laboratory frame

The component V(2,k) of the second-rank spherical tensor V in the obs coordinate frame is:

EFG tensor component V(2,k) Equation 2

The Wigner matrix for passive rotation is:

EFG tensor component V(2,k)

Reference

  1. J. Van de Wiele
    Rotations et moments angulaires en mécanique quantique,
    Ann. Phys. Fr. 26, 1-169 (2001).
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  2. D. Freude
    Quadrupolar nuclei in solid-state nuclear magnetic resonance (PDF article),
    in Encyclopedia of Analytical Chemistry
    R. A. Meyers Ed.
    J. Wiley, Chichester, Pages 12188-12224 (2000).
     
  3. B. F. Chmelka and J. W. Zwanziger
    Solid-state NMR line narrowing methods for quadrupolar nuclei: Double rotation and dynamic-angle spinning,
    in NMR Basic Principles and Progress, Vol. 33, Pages 79-124
    P. Diehl, E. Fluck, H. Günther, R. Kosfeld, and J. Seelig Eds.
    B. Blümich (Guest Editor)
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  4. D. Freude and J. Haase
    Quadrupole effects in solid-state nuclear magnetic resonance,
    in NMR Basic Principles and Progress, Vol. 29, Pages 1-90
    P. Diehl, E. Fluck, H. Günther, R. Kosfeld, and J. Seelig Eds.
    Springer, Berlin (1993).
     
  5. G. Engelhardt and D. Michel
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  6. Karl Blum
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    Plenum, New York (1981).
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  7. Mitchel Weissbluth
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    Academic, New York (1978).

    The Wigner passive rotation matrix presented in this book is D(2, passive,W)(α, β, γ).

     
  8. Ulrich Haeberlen
    High Resolution NMR in Solids, Selective Averaging,
    Academic, New York (1976), Pages 11-12.

    The three angles α, β, and γ are the Euler angles by which the laboratory frame can be brought into coincidence with the principal axis system.

     
  9. A. R. Edmonds
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    Princeton University Press, Princeton (1977).

    In the following discussions the term rotation will be interpretated as a rotation of the frame of reference about the origin, the field points (i.e. the physical system) being supposed fixe. Each point of the 3-dimensional space is thus given new coordinates, which are functions of the old coordinates and of the parameters which describe the rotation, namely the Euler angles.

     
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    Abstract
     

C. Conclusion

Both Equations 1 and 2 represent a passive rotation of the EFG tensor:
- from the PAS coordinate frame to the obs coordinate frame for Equation 1,
- from the obs coordinate frame to the PAS coordinate frame for Equation 2.

The Euler angles α, β, and γ in Equation 1 correspond to the Euler angles γ, β, and α in Equation 2, respectively.

Sometimes, Equation 2 is written in the following way:

EFG tensor component V(2,k) Equation 3

The matrix D(2,passive,W)(α, -β, γ) in Equation 3 is the transpose of D(2,passive,W)(γ, β, α) defined in Equation 2.

Reference

  1. Rytis Juršėnas and Gintaras Merkelis
    The transformation of irreducible tensor operators under spherical functions,
    Int. J. Theor. Phys. 49, 2230-2245 (2010).
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  2. Jeffrey W. Peng and Gerhard Wagner
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    in Nuclear Magnetic Resonance Probes of Molecular Dynamics, Pages 373-454
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  3. Eugene P. Wigner
    Group Theory and Its Application to Quantum Mechanics of Atomic Spectra Group Theory and Its Application to Quantum Mechanics of Atomic Spectra, Volume 5 in Pure and Applied Physics,
    translated from the Germain by J. J. Griffin
    Academic Press, New York (1959), Pages 168-169, 357-359, and 362.
    5 Mb DjVu file, DjVu viewer from DjVu.org, Plug-in DjVu test page

     
 

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