Extending numerical simulations in SIMPSON: Electron paramagnetic resonance, dynamic nuclear polarisation, propagator splitting, pulse transients, and quadrupolar cross terms

Abstract

Aimed at the simulation, design, and interpretation of advanced pulse experiments crossing the boundaries between nuclear magnetic resonance (NMR) and electron paramagnetic resonance (EPR), including the rapidly emerging, hybrid discipline of pulsed dynamic nuclear polarisation (DNP), we present a host of novel features in the widely used SIMPSON software package addressing these aspects. Along with this come new features for advanced pulse sequence evaluation in terms of propagator splitting, high-order spin operator cross terms, and pulse phase transients. These fundamental new tools are introduced in a C++-based next generation of the SIMPSON software, which improves calculations speed in some aspects, is better prepared for further developments, and facilitates easier community contributions to the open-source software package.

Figures (9)

• Figure 1: Examples of visualisation of and data integration with simpson simulations. (A) The easynmr workflow with three different simpson simulations. The left frame shows the three simulation objects, each forwarding data to a plotting object. The right frame shows the content of each object, here highlighting the plot of one of the simulations. (B) An example of SimView, showing the layout of the interface: a text editor where the simpson input file can be modified; a text output from a running simpson calculation; and a graphical area where calculated fids or spectra can be plotted.
• Figure 2: (A) Pulse sequence diagrams for 2-pulse and 3-pulse eseem experiments. (B) Simulated echo intensity, $\langle \hat{\mathrm{S}}_x \rangle$), as a function of the delay time $\tau$ for 2-pulse eseem for a single orientation of an electron-proton two-spin system with ideal pulses (using pulseid). (C) Corresponding simulation for 3-pulse eseem using the same spin system, but with a summation over several crystallites. Additionally, an explicit summation over electron offsets is performed. The effect of using non-ideal (soft) pulses (red, using pulse) is shown relative to ideal-pulses (blue, using pulseid).
• Figure 3: (A) Pulse diagrams for representative pulsed dnp experiments. (B,C) Numerical simulations of the $\text{e}^{-}$ to $^{1}\text{H}$ polarisation transfer efficiency $|\langle \hat{\mathrm{I}}_z\rangle|$ (numerical) as a function of the electron-spin offset $\Delta \omega_S/(2\pi)$ (represented by the simpson parameter gtensor_1_iso) (B) and the contact (or mixing) time $t_{\mathrm{contact}}$. The initial operator (start_operator) is $\hat{\mathrm{S}}_x$; this assumes, as in the pulse diagrams, initialisation with an ideal $(\pi/2)_y$ pulse (not included in the simulation). The detection operator (detect_operator) is along $\hat{\mathrm{I}}_z$, which assumes detection of transverse coherence with a solid-echo pulse sequence (not included in the simulation). The pulse sequence diagrams also contain saturation pulses and repetition of the dnp element to pump polarisation from the electron to multiple nuclear spins, which has not been included in the two-spin simulations. The simpson code and the spin-system parameters for these figures are outlined in the text and provided in the Supplementary Material.
• Figure 4: Gradient-free optimisation of a composite inversion pulse to mitigate large RF-field inhomogeneity. A shaped pulse with a duration of $125~µs$ comprising five elements, was optimised using the simplex algorithm and the prop_split method; the resulting RF amplitudes $\omega_\mathrm{RF}/(2\pi)$ and phases $\phi$ are presented in panels (A) and (B), respectively. (C) The fidelity $F_{\!\mathrm{s}}$ of the inversion is evaluated over a range of RF-field scaling factors $\kappa_\mathrm{RF}$ and compared with the inversion profile of a rectangular pulse with a $20~kHz$ amplitude.
• Figure 5: (A) Plot of the constant-amplitude ($10~kHz$) sordor pulse Goodwin2020, with $Q=0.84$ and $F_{\!\mathrm{u}}=99.095%$, optimised using the simpson code outlined in the text. (B) Wall-clock time for fidelities achieved with morphic optimal control, starting from burbopKobzar2012 and ramping the quadratic coefficient, $Q$ (burbop has $Q=0$), for both method diag and method prop_split; dashed lines represent an optimisation of phase and amplitude, while solid lines represent phase-only optimisation. Wall-clock time were measured using a Linux workstation with an amd Ryzen 7 5700G 8-core processor.