Simulation of pulsed dynamic nuclear polarization in the steady state

n pulsed dynamic nuclear polarization (DNP), enhancement of bulk nuclear polarization requires the repeated application of a microwave
pulse sequence. So far, analysis of a one-time transfer of electron spin polarization to a dipolar-coupled nuclear spin has guided the design of
DNP pulse sequences. This has obvious shortcomings, such as the inability to predict the optimal repetition time. In an actual pulsed DNP
experiment, a balance is reached between the polarization arriving from the unpaired electrons and nuclear relaxation. In this article, we
explore three algorithms to compute this stroboscopic steady state: (1) explicit time evolution by propagator squaring, (2) generation of an
effective propagator using the matrix logarithm, and (3) direct calculation of the steady state with the Newton–Raphson method. Algorithm
(2) is numerically unstable in dissipative DNP settings. Algorithms (1) and (3) are both stable; algorithm (3) is the most efficient. We compare
the steady-state simulations to existing experimental results at 0.34 and 1.2 T and to the first experimental observation of X-inverse-X (XiX)
DNP at 3.4 T. The agreement is good and improves further when electron–proton distance and electron Rabi frequency distributions are
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accounted for. We demonstrate that the trajectory of the spin system during one-time application of a microwave pulse sequence differs from
the steady orbit. This has implications for DNP pulse sequence design.