Figure 1

Sample ${\mathrm{C}}_{60}$. (a) Comparison of CPMG (gray) to APCPMG (blue). Inserting a single flip-${180}_Y$ pulse into APCPMG induces an echo of the echo train (green). (b) Inserting a single ${180}_X$ (red) has no effect (blue). (c) Reversing the APCPMG phase pattern ${90}_X-\{-Y,Y{\}}^{200}-\{Y,-Y{\}}^{600}$ at the point indicated has the same effect (black) as inserting a single ${180}_Y$ pulse (green). (d) A CPMG of the echo train is induced by using ${90}_X-\{-Y,Y{\}}^{10}-(\{Y,-Y{\}}^{20}-\{-Y,Y{\}}^{20}{)}_{\mathrm{repeat}}$. For (a)–(d), $\tau =25\mu \mathrm{s}$, ${\Omega }_{\mathrm{offset}}^{\mathrm{global}}=0$, $\alpha \approx 0.71$, and only the peak of each echo is shown. The signals in Figs. 1, 2, 3, 4 are normalized to the amplitude of the ${\mathrm{C}}_{60}$ or Si:Sb FID signal.Reuse & Permissions

Figure 2

Sample ${\mathrm{C}}_{60}$. Three experiments inspired by the magic echo [10, 11], which all start with $\{-X,X{\}}^N$, have distinctly different results. With ${90}_{-X}$ following the repeating block, no magic echo forms (red); with ${90}_{+X}$ following the repeating block, a large echo emerges (blue); when applying a ${180}_Y$ pulse at time $t_{f_1}$ after the burst of the failed sequence (red), an optimized echo is achieved (green). Here $N=200$, $\tau =50\mu \mathrm{s}$, ${\Omega }_{\mathrm{offset}}^{\mathrm{global}}=0$, and $\alpha \approx 0.83$.Reuse & Permissions

Figure 3

Sample ${\mathrm{C}}_{60}$. Using ${\Omega }_z^{\mathrm{net},\pm }={\Omega }_z^{\mathrm{loc}}\pm {\Omega }_{\mathrm{offset}}^{\mathrm{global}}$, the quadratic echoes produced by $\{X,X{\}}^{N/2}\mathrm{\text{−}}\{-X,-X{\}}^{N/2}-{90}_Y-t_{\mathrm{free}}$, green (${\nu }_{\mathrm{offset}}=0\mathrm{Hz}$) and black (${\nu }_{\mathrm{offset}}=-3\mathrm{kHz}$), differ from the linear echoes produced by $\{-X,X{\}}^{N/2}\{X,-X{\}}^{N/2}-{90}_X-t_{\mathrm{free}}$, blue (${\nu }_{\mathrm{offset}}=0\mathrm{Hz}$) and red (${\nu }_{\mathrm{offset}}=-1\mathrm{kHz}$), where ${\Omega }_{\mathrm{offset}}^{\mathrm{global}}=-h{\nu }_{\mathrm{offset}}$. Only the black echo shifts to the right. Inset: Image plot of 31 quadratic echoes for $0\mathrm{Hz}\le {\Omega }_{\mathrm{offset}}^{\mathrm{global}}/h\le 3\mathrm{kHz}$, in steps of 100 Hz. The black trend line shows our predicted Zeeman refocusing time. Here $N=100$, $\tau =10\mu \mathrm{s}$, and $\alpha \approx 0.5$.Reuse & Permissions

Figure 4

(a) Sample Si:Sb. The ${}^{29}\mathrm{Si}$ time-suspension data using the sequence ${90}_X-\{2,0,-Y,-Y{\}}^{84000}$, with $\tau =60\mu \mathrm{s}$, ${\nu }_{\mathrm{offset}}=2.5\mathrm{kHz}$ (blue), and corresponding fitting curve (black), extend far beyond the normal ${}^{29}\mathrm{Si}$ FID with ${\nu }_{\mathrm{offset}}=0\mathrm{Hz}$ (red). (a, inset) The 200 Hz normal spectrum (red) is narrowed to 0.003 Hz (black, Fourier transformation of the fitting curve), centered at ${\nu }_{\mathrm{offset}}$. (b) Sample ${\mathrm{C}}_{60}$. Reproduction of a top-hat line shape using sequence ${90}_X-\{2,t_0,-Y,-Y\}-\{2,0,-Y,Y{\}}^{30}$, with $\tau =22\mu \mathrm{s}$ and $t_0=0$. Each trace is the measured spectrum of a pseudo-FID with different ${\nu }_{\mathrm{offset}}$, for $-4\mathrm{kHz}\le {\nu }_{\mathrm{offset}}\le +4\mathrm{kHz}$ in steps of 500 Hz, covering the range $2\pi |{\nu }_{\mathrm{offset}}|/{\omega }_1\le 16\%$. To obtain this full bandwidth, the pseudo-FID interleaves a second data set using the same sequence but with $t_0=-(\frac{\Delta }{2}+\frac{1}{2{\omega }_1})$.Reuse & Permissions