Spin dynamics in the pulsed spin locking of nuclear quadrupole resonance

We theoretically examine the perturbative effects of a series of radio-frequency (rf) pulses, electric field gradient inhomogeneity, and dipole-dipole coupling on the spin dynamics of spin-1 nuclei dominated by the quadrupole interaction. The dipole-dipole coupling is between neighboring spin-1 nuclei with identical nuclear quadrupole resonance frequencies, but the principal axes frames of the electric field gradient at each nucleus are not aligned. Such a comprehensive treatment is necessary to determine the optimal sequence of rf pulses which maximizes the echoes in the detection of the substance of interest, for example, an explosive or other contraband material. We confirm our theoretical model using the nitrogen in powder samples of $p$-chloroaniline and sodium nitrite.

Figure 1

(Color online) The energy levels for two nitrogen atoms are labeled using the notation $\mid ab⟩$, corresponding to the two spins $a$ and $b$ each with one of the possible eigenstates of $H_Q$ ($\mid 0⟩$, ∣−⟩, or ∣+⟩). The nonzero matrix elements of the dipolar Hamiltonian and the rf Hamiltonian in the interaction representation of the quadrupole Hamiltonian give the possible transitions. In the figure, the degenerate levels are separated to show the flip-flop terms of the dipolar Hamiltonian. The gray levels, or $V$-levels, form a subsystem isolated from the black levels, or $W$-levels.

Figure 2

The orientation of the rf excitation $({\theta }_L,{\varphi }_L)$ with respect to the combined PAS frames of neighboring nuclei, defined by the average of the relevant axes ${\widehat{y}}_{\mathit{avg}}$ and the difference ${\widehat{y}}_{\mathit{dif}}$.

Figure 3

(Color online) (a) The function $g(N,\theta ,p)$ characteristic of the $N\text{th}$ echo size for an effective refocusing pulse $\theta$ in the presence of dipolar coupling and (b) how the average value of $g$ increasingly deviates from unity for increasing dipolar coupling or $p$. (c)–(e) illustrate for a powder sample with $\eta =\xi =\zeta =10°$ the relative contributions to the average echo size for increasing dislocation between PAS frames of neighboring nuclei by ${\theta }_{EFG}$. Notice how the maxima and minima occur at increasing refocusing pulse strengths as ${\theta }_{EFG}$ increases.

Figure 4

Considering only EFGs and delta-function pulses (a) for a single crystal and (b) for a powder sample, shows the $N\text{th}$ echo size (units of $\frac{2\hslash {\omega }_y}{3kT}$), for a Gaussian linewidth (full width at half $\text{maximum}=150\mathrm{Hz}$) and $\tau =1\mathrm{ms}$ as a function of the refocusing pulse strength. (c) for a single crystal and (d) for a powder sample shows how the average signal size (for 40 echoes) varies little as a function of off-resonance for the optimal refocusing pulse, but quite significantly for refocusing pulses of half that value.

Figure 5

For sodium nitrite, irradiated at ${\omega }_z$ $\left(1.038\mathrm{MHz}\right)$ and using an optimum preparatory pulse of 119°, the signal from the first echo (theory in gray dashed line, experiment in gray circles) is compared to the average echo of 32 (theory in black solid line, experiment in black squares) as a function of refocusing pulse strength. The double peak of the averaged data is characteristic of a sample whose linewidth is dominated by dipolar coupling. For an optimal refocusing pulse of 150°, we also varied the preparatory pulse strength (theory in black dotted line, experiment in crosses). For all cases, $\tau =1.9\mathrm{ms}$, pulse durations are $t_0=120\mu \mathrm{s}$ and $t_r=150\mu \mathrm{s}$, and signals are normalized with respect to the optimal average echo.

Figure 6

(Color online) For $p$-chloroaniline irradiated at (a) ${\omega }_x$ $\left(3.263\mathrm{MHz}\right)$ or (b) ${\omega }_y$ $\left(2.767\mathrm{MHz}\right)$, using an optimum preparatory pulse of 119°, the average of 32 echoes varies as a function of pulse spacing $\tau$ and as a function of the refocusing pulse strength (theory in solid lines, experiments in filled symbols). The field strength was calibrated by fixing the refocusing pulse at 150° and varying the preparatory pulse (theory in dotted lines, experiments in open symbols). Pulse durations were $t_0=160\mu \mathrm{s}$ and $t_r=200\mu \mathrm{s}$ and signals for different $\tau$’s are shown normalized to their respective calibration curves.

Figure 7

For a sample of $p$-chloroaniline, irradiated at ${\omega }_y$, as a function of off-resonance $[\Delta \omega ∕\left(2\pi \right)]$ for two different pulse spacing. Theory is shown as lines and the data as symbols, with the grid lines spaced at $1∕(2\tau +t_r)$ to correspond to the predicted maxima. An optimum preparatory pulse of 119° and a refocusing pulse of 150° was used with pulse durations of $t_0=80\mu \mathrm{s}$ and $t_r=100\mu \mathrm{s}$.