We report the properties of more than 800 bursts detected from the repeating fast radio burst (FRB) source FRB 20201124A with the Five-hundred-meter Aperture Spherical radio Telescope during an extremely active episode on UTC 2021 September 25th-28th in a series of four papers. In this fourth paper of the series, we present a systematic search of the spin period and linear acceleration of the source object from both 996 individual pulse peaks and the dedispersed time series. No credible spin period was found from this data set. We rule out the presence of significant periodicity in the range between 1 ms and 100 s with a pulse duty cycle <0.49 ± 0.08 (when the profile is defined by a von-Mises function, not a boxcar function) and linear acceleration up to 300 m s in each of the four one-hour observing sessions, and up to 0.6 m s in all 4 days. These searches contest theoretical scenarios involving a 1 ms–100 s isolated magnetar/pulsar with surface magnetic field <10 G and a small duty cycle (such as in a polar-cap emission mode) or a pulsar with a companion star or black hole up to 100 M and P > 10 hr. We also perform a periodicity search of the fine structures and identify 53 unrelated millisecond-timescale "periods" in multi-components with the highest significance of 3.9σ. The "periods" recovered from the fine structures are neither consistent nor harmonically related. Thus they are not likely to come from a spin period. We caution against claiming spin periodicity with significance below ∼4σ with multi-components from one-off FRBs. We discuss the implications of our results and the possible connections between FRB multi-components and pulsar microstructures.
: (stars:) pulsars: general – methods: observational – radio continuum: general
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