Using a sample of 208 broad-line active galactic nuclei (AGNs) from Swift/BAT AGN Spectroscopic Survey in ultra-hard X-ray band (14 − 195 keV), the hot corona properties are investigated, i.e. the fraction of gravitational energy dissipated in the hot corona and the hard X-ray photon index. The bolometric luminosity, L Bol , is calculated from host-corrected luminosity at 5100Å. Virial supermassive black hole masses (SMBH, M BH ) are calculated from the Hβ line width and the corresponding broad line region size-luminosity empirical relation at 5100Å. We find a strong anti-correlation between the fraction of energy released in corona (F X ≡ L 14−195keV /L Bol ) and the Eddington ratio (ε ≡ L Bol /L Edd ), F X ∝ ε −0.60±0.1 . It is found that this fraction also has a correlation with the SMBH mass,
BH. Assuming that magnetic buoyancy and feild reconnection lead to the formation of a hot corona, our result favours the shear stress tensor being a proportion of the gas pressure. For our entire sample, it is found that the hard X-ray photon index Γ has a weak but significant correlation with the Eddington ratio, Γ = 2.17 + 0.21 log ε. However, this correlation is not robust because the relation is not statistically significant for its subsample of 32 RM AGNs with relatively reliable M BH or its subsample of 166 AGNs with singleepoch M BH . We do not find a statistically significant relation between the photon index and the Eddington ratio taking into account an additional dependence on F X . disk. Therefore, the fraction of FX ≡ LX/L Bol (LX and L Bol are the X-ray luminosity and the bolometric luminosity respectively) can be obtained if the magnetic stress and energy transportation are assumed (e.g. Merloni & Fabian 2002). This may give us a possible opportunity to test the working magnetic stress from hard X-ray observations. The relation between the fraction of energy dissipated in the corona FX and the Eddington ratio ε (ε ≡ L Bol /L Edd , L Edd is the Eddington luminosity) was investigated by some authors (e.g. Merloni & Fabian 2002;Wang et al. 2004;Yang et al. 2006). For a compiled sample of 56 AGNs from ASCA observation, Wang et al. (2004) found a relation between FX and ε as FX ∝ ε −0.64±0.09 , where the X-ray luminosity in 2 − 10 keV is used. Considering a larger sample of 98 AGNs, Yang et al. (2006) found FX ∝ ε −0.66 . These results supported the magnetic stress tensor being the form of t rφ ∝ Pgas, where Pgas is the gas pressure. In their compiled sample, they only have nine sources with ultra-hard X-ray observations of INTEGRAL and Swift, and for other sources the luminosity in 2 − 150 kev is extrapolated from the 2-10 keV luminosity with a fixed photon index. A sample of more AGNs with direct harder X-ray than 2-10 keV is needed for further investigation, such as Swift/BAT. Another relation between the X-ray photon index Γ and c 0000 The Authors