Symmetry Pathways in Solid-State NMR

Symmetry Pathways in Solid-State NMR

P. J. Grandinetti, J. T. Ash, and N. M. Trease
065 - Prog. NMR Spect., 59, 121-196 (2011). (originally published)

Abstract

In this review we have outlined a simple and consistent framework for designing NMR experiments, particularly for solid-state NMR. This framework extends the concept of coherence transfer pathways, starting with two main pathways called the spatial pathway and the spin transition pathway which completely describe an NMR experiment. Given a pulse sequence and spin system's spatial and spin transition pathways a series of related symmetry pathways can be derived which show, at a glance, when and which frequency components for the system will refocus into echoes. Although these frequency components are classified according to familiar symmetries under the orthogonal rotation subgroup, (i.e., $s$, $p$, $d$, $f$, $\ldots$), the power of this framework is in providing insight behind many experiments even when internal couplings are much larger than the rf coupling and one can no longer rely on the symmetries under the orthogonal rotation subgroup as a guide to designing new experiments. Additionally, this framework provides a more physical picture behind the use of affine transformations when processing the multidimensional signals obtained in many solid-state NMR experiments, and also serves as a useful guide when designing multi-dimensional NMR experiments with pure absorption mode lineshapes.

This framework not only provides a powerful tool for designing new NMR experiments, but can be a useful pedagogical tool for NMR, allowing students to quickly grasp a number of modern solid-state NMR experiments without the need to enter into a full density operator description of each experiment.

Notes

In an updated edition of our review article (4 October 2013), a small error in the caption of Fig. 2 has been corrected, and a few notations, used throughout the published version, have been modified in this edition such that,

  1. individual states are no longer enclosed in angle brackets, i.e., $m_i$ instead of $\langle m_i \rangle$,
  2. a transition from state $i$ to $j$ is represented by $| j \rangle \langle i |$, instead of $\left(\langle i \rangle, \langle j \rangle \right)$,
  3. single transition Zeeman order associated with states $i$ and $j$ is represented by $[i,j]$, instead of $[\langle i \rangle, \langle j\rangle]$.
  4. the $l$th-rank spin transition symmetry function of the $k$th frequency component is represented as ${\xi}_l^{(k)}(i,j)$, instead of ${\xi}_k(i,j)$
  5. the $L$th-rank orientational spatial symmetry function of the $k$th frequency component is represented as ${\Xi}_L^{(k)}(\Theta )$, instead of ${\Xi}_k(\Theta )$.
  6. an unnecessary scaling factor in Eqs. (105) and (106) was eliminated.
  7. the sign of effective $\langle D_n \cdot p_I\rangle$ evolution during $\epsilon$ in Fig. 38 has been reversed to be consistent with positive going echo associated with refocusing of rotor modulated anisotropic evolution.
  8. an incorrect factor of $1/\sqrt{2}$ instead of $\sqrt{\frac{3}{2}}$ was corrected in Eq. (A.185).
  9. an incorrect factor of appeared in the first-order nuclear shielding proportionality constant. It now shows the correct value which was already given in Eq. (A.191).
  10. an incorrect factor of $1/\sqrt{2}$ instead of $\sqrt{\frac{3}{2}}$ was corrected in Eq. (A.222).
  11. the zero-rank first-order proportionality constant for the strong $J$ coupling was eliminated since the spin transition part of the zero-rank term is zero.
  12. an incorrect factor of $\sqrt{3}$ in the first-order strong $J$ coupling proportionality constant given in Eq. (A.227) was eliminated.
  13. an incorrect factor of appeared in the first-order strong $J$ coupling proportionality constant. It now shows the correct value which was already given in Eq. (A.229).
  14. an incorrect factor of appeared in the first-order weak $J$ coupling proportionality constant. It now shows the correct value which was already given in Eq. (A.238).
  15. an incorrect factor of $\sqrt{\frac{2}{3}}$ instead of $\sqrt{\frac{3}{2}}$ was corrected in Eq. (A.149).
  16. the symbols for the spherical tensor components of the dipolar coupling were mislabeled in Eqs. (A.264)-(A.266).
  17. the right hand side of Eq. (A.269) should have been $-\displaystyle \frac{\mu_0}{2 \pi} \zeta_d \gamma_1 \gamma_2 \hbar$.
  18. an incorrect factor of $1/4$ in Eqs. (A.270), (A.271), (A.273), (A.274), (A.276), (A.280), (A.281), (A.283), (A.284), (A.285) was eliminated, with corresponding changes for $d_{II}$ and $d_{IS}$ in Table I.
  19. a missing factor of $1/2$ was added to Eqs. (A.288)-(A.290) and Eqs. (A.292)-(A.294) with corresponding changes for $d_{II}$ and $d_{IS}$ in Table I.

Additionally, the manuscript has been reformatted as single column to improve readability and minimize line breaks in equations. Below are the updated input files for the manuscript.

  1. 01-FrequencySum.pdf
  2. 02-TransitionDiagram.pdf
  3. 03-TransitionDiagramPopulations.pdf
  4. 04-MASPathway.pdf
  5. 05-pValues.pdf
  6. 06-dValues.pdf
  7. 07-fValues1.pdf
  8. 08-fValues2.pdf
  9. 09-p32p12Coupled.pdf
  10. 10-d32p12Coupled.pdf
  11. 11-p32p32Coupled.pdf
  12. 12-d32p32Coupled.pdf
  13. 13-p32d32Coupled.pdf
  14. 14-ValuesSpin1.pdf
  15. 15-ValuesSpin32.pdf
  16. 16-ValuesSpin52.pdf
  17. 17-ValuesSpin3.pdf
  18. 18-ValuesSpin72.pdf
  19. 19-ValuesSpin92-c0.pdf
  20. 20-ValuesSpin92-c2.pdf
  21. 21-ValuesSpin92-c4.pdf
  22. 22-SpinOnePathways.pdf
  23. 23-SpinOneBloch.pdf
  24. 24-TwoSpinBlochSpectrum.pdf
  25. 25-HahnEcho.pdf
  26. 26-HahnEchoSpin1.pdf
  27. 27-SEDOR.pdf
  28. 28-DoubleQuantumEchoSpinOne.pdf
  29. 29-SolidEcho.pdf
  30. 30-StimulatedSolidEcho.pdf
  31. 31-SolomonEchoes.pdf
  32. 32-HETCOR.pdf
  33. 33-HSQC.pdf
  34. 34-DnEcho.pdf
  35. 35-TOP.pdf
  36. 36-TOP1.pdf
  37. 37-TOP2.pdf
  38. 38-2DPASS.pdf
  39. 39-DOR.pdf
  40. 40-MQ-MASEcho.pdf
  41. 41-MQ-MASShear.pdf
  42. 42-ST-MASEcho.pdf
  43. 43-DQFST-MASEcho.pdf
  44. 44-COASTEREchoes.pdf
  45. 45-COASTERShear.pdf
  46. 46-COASTERShear2.pdf
  47. 47-etasmall.pdf
  48. 48-RelativeGS.pdf
  49. 49-MQDOREchoes.pdf
  50. 50-MAF.pdf
  51. 51-VACSY.pdf
  52. 52-DASEcho.pdf
  53. 53-p2p4time.pdf
  54. 54-1dfid.pdf
  55. 55-1decho.pdf
  56. 56-TotalEchoSpin1.pdf
  57. 57-2Dlineshapes.pdf
  58. 58-PureAbsorptionModeAcquisition.pdf
  59. 59-2DExchangepathway+-.pdf
  60. 60-Hypercomplex.pdf
  61. 61-2DShiftedEchopathway.pdf
  62. 62-ShiftedEcho.pdf
  63. 63-TDTrans.pdf
  64. 64-Shear_t2.pdf
  65. 65-Shear_t1.pdf
  66. 66-DigitalYShear.pdf
  67. 67-SingleShearGray.pdf
  68. 68-DoubleShearGray.pdf
  69. 69-SidebandShearing.pdf
  70. 70-GroupTheory.pdf
  71. 71-MASDASTrajectory.pdf
  72. 72-IterativeTrajectory.pdf
  73. 065-JPNMRS_59_121_2011_2ndEd.tex