We report on the results of detailed X-ray spectroscopy of the Fe K region in the Seyfert 1 galaxy NGC 3783 from five ~170 ks observations with the Chandra high energy gratings. Monitoring was conducted over an interval of ~125 days in 2001. The combined data constitute the highest signal-to-noise Fe K spectrum having the best velocity resolution in the Fe K band to date (FWHM ~1860 km/s). The data show a resolved Fe K line core with a center energy of 6.397 +/- 0.003 keV, consistent with an origin in neutral or lowly ionized Fe, located between the BLR and NLR, as found by Kaspi et al. (2002). We also find that excess flux around the base of the Fe K line core can be modeled with either a Compton scattering ``shoulder or an emission line from a relativistic accretion disk, having an inclination angle of 11 degrees or less. This disk line model is as good as a Compton-shoulder model for the base of the Fe K line core. In the latter model, the column density is 7.5 [+2.7,-0.6] x 10^{23} cm^{-2}, which corresponds to a Thomson optical depth of ~0.60. An intrinsic width of 1500 [+460,-340] km/s FWHM is still required in this model. Moreover, more complicated scenarios involving both a Compton-shoulder and a disk line cannot be ruled out. We confirm an absorption feature due to He-like Fe (FWHM = 6405 [+5020,-2670] km/s), found in previous studies.
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