We present paper six of the NIRSPEC Brown Dwarf Spectroscopic Survey, an analysis of multi-epoch, high-resolution (R~20,000) spectra of 25 field dwarf systems (3 late-type M dwarfs, 16 L dwarfs, and 6 T dwarfs) taken with the NIRSPEC infrared spectrograph at the W. M. Keck Observatory. With a radial velocity precision of ~2 km/s, we are sensitive to brown dwarf companions in orbits with periods of a few years or less given a mass ratio of 0.5 or greater. We do not detect any spectroscopic binary brown dwarfs in the sample. Given our target properties, and the frequency and cadence of observations, we use a Monte Carlo simulation to determine the detection probability of our sample. Even with a null detection result, our 1 sigma upper limit for very low mass binary frequency is 18%. Our targets included 7 known, wide brown dwarf binary systems. No significant radial velocity variability was measured in our multi-epoch observations of these systems, even for those pairs for which our data spanned a significant fraction of the orbital period. Specialized techniques are required to reach the high precisions sensitive to motion in orbits of very low-mass systems. For eight objects, including six T dwarfs, we present the first published high-resolution spectra, many with high signal to noise, that will provide valuable comparison data for models of brown dwarf atmospheres.
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