Aims: We present a spectral analysis of a new, flux-limited sample of 72 X-ray selected clusters of galaxies identified with the X-ray Telescope (XRT) on board the Swift satellite down to a flux limit of ~10-14 erg s-1 cm-2 (SWXCS). We carry out a detailed X-ray spectral analysis with the twofold aim of measuring redshifts and characterizing the properties of the intracluster medium (ICM) for the majority of the SWXCS sources. Methods: Optical counterparts and spectroscopic or photometric redshifts for some of the sources are obtained with a cross-correlation with the NASA/IPAC Extragalactic Database. Additional photometric redshifts are computed with a dedicated follow-up program with the Telescopio Nazionale Galileo and a cross-correlation with the SDSS. In addition, we also blindly search for the Hydrogen-like and He-like iron Kα emission line complex in the X-ray spectrum. We detect the iron emission lines in 35% of the sample, and hence obtain a robust measure of the X-ray redshift zX with typical rms error 1-5%. We use zX whenever the optical redshift is not available. Finally, for all the sources with measured redshift, background-subtracted spectra are fitted with a single-temperature mekal model to measure global temperature, X-ray luminosity and iron abundance of the ICM. We perform extensive spectral simulations to accounts for fitting bias, and to assess the robustness of our results. We derive a criterion to select reliable best-fit models and an empirical formula to account for fitting bias. The bias-corrected values are then used to investigate the scaling properties of the X-ray observables. Results: Overall, we are able to characterize the ICM of 46 sources with redshifts (64% of the sample). The sample is mostly constituted by clusters with temperatures between 3 and 10 keV, plus 14 low-mass clusters and groups with temperatures below 3 keV. The redshift distribution peaks around z ~ 0.25 and extends up to z ~ 1, with 60% of the sample at 0.1 < z < 0.4. We derive the luminosity-temperature relation for these 46 sources, finding good agreement with previous studies. Conclusions: Thanks to the good X-ray spectral quality and the low background of Swift/XRT, we are able to measure ICM temperatures and X-ray luminosities for the 46 sources with redshifts. Once redshifts are available for the remaining 26 sources, this sample will constitute a well-characterized, flux-limited catalog of clusters distributed over a broad redshift range (0.1 ≤ z ≤ 1.0) providing a statistically complete view of the cluster population with a selection function that allows a proper treatment of any measurement bias. The quality of the SWXCS sample is comparable to other samples available in the literature and obtained with much larger X-ray telescopes. Our results have interesting implications for the design of future X-ray survey telescopes, characterized by good-quality PSF over the entire field of view and low background. Tables 1 and 2 and Appendix A are available in electronic form at http://www.aanda.orgCatalog and data products of SWXCS, constantly updated, are made available to the public through the websites http://www.arcetri.astro.it/SWXCS/ and http://swxcs.ustc.edu.cn/
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