# Symmetry Pathways in Solid-State NMR

P. J. Grandinetti, J. T. Ash, and N. M. Trease
065 - Prog. NMR Spect., 59, 121-196 (2011). (doi: 10.1016/j.pnmrs.2010.11.003)

# 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, 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.