The transformation stretch tensor plays an essential role in the evaluation of conditions of compatibility between phases and the use of the Cauchy-Born rule. This tensor is difficult to measure directly from experiment. We give an algorithm for the
determination of the transformation stretch tensor from x-ray measurements of structure and lattice parameters. When evaluated on some traditional and emerging phase transformations the algorithm gives unexpected results.
Off-center stellar tidal disruption flares have been suggested to be a powerful probe of recoiling supermassive black holes (SMBHs) out of galactic centers due to anisotropic gravitational wave radiations. However, off-center tidal flares can also be
produced by SMBHs in merging galaxies. In this paper, we computed the tidal flare rates by dual SMBHs in two merging galaxies before the SMBHs become self-gravitationally bounded. We employ an analytical model to calculate the tidal loss-cone feeding rates for both SMBHs, taking into account two-body relaxation of stars, tidal perturbations by the companion galaxy, and chaotic stellar orbits in triaxial gravitational potential. We show that for typical SMBHs with mass 10^7 M_sun, the loss-cone feeding rates are enhanced by mergers up to Gamma ~ 10^{-2} yr^{-1}, about two order of magnitude higher than those by single SMBHs in isolated galaxies and about four orders of magnitude higher than those by recoiling SMBHs. The enhancements are mainly due to tidal perturbations by the companion galaxy. We suggest that off-center tidal flares are overwhelmed by those from merging galaxies, making the identification of recoiling SMBHs challenging. Based on the calculated rates, we estimate the relative contributions of tidal flare events by single, binary, and dual SMBH systems during cosmic time. Our calculations show that the off-center tidal disruption flares by un-bound SMBHs in merging galaxies contribute a fraction comparable to that by single SMBHs in isolated galaxies. We conclude that off-center tidal disruptions are powerful tracers of the merging history of galaxies and SMBHs.
Supermassive black hole binaries (SMBHBs) are expected by the hierarchical galaxy formation model in $Lambda$CDM cosmology. There is some evidence in the literature for SMBHBs in AGNs, but there are few observational constraints on the evolution of S
MBHBs in inactive galaxies and gas-poor mergers. On the theoretical front, it is unclear how long is needed for a SMBHB in a typical galaxy to coalesce. In this paper we investigate the tidal interaction between stars and binary BHs and calculate the tidal disruption rates of stellar objects by the BH components of binary. We derive the interaction cross sections between SMBHBs and stars from intensive numerical scattering experiments with particle number $sim10^7$ and calculate the tidal disruption rates by both single and binary BHs for a sample of realistic galaxy models, taking into account the general relativistic effect and the loss cone refilling because of two-body interaction. We estimate the frequency of tidal flares for different types of galaxies using the BH mass function in the literature. We find that because of the three-body slingshot effect, the tidal disruption rate in SMBHB system is more than one order of magnitude smaller than that in single SMBH system. The difference is more significant in less massive galaxies and does not depend on detailed stellar dynamical processes. Our calculations suggest that comparisons of the calculated tidal disruption rates for both single and binary BHs and the surveys of X-ray or UV flares at galactic centers could tell us whether most SMBHs in nearby galaxies are single and whether the SMBHBs formed in gas-poor galaxy mergers coalesce rapidly.
