Surface geometry and shadow of a Schwarzschild black hole with a halo

We studied the surface geometry and shadows of a Schwarzschild black hole with a halo containing quadrupolar and octopolar terms.It was found that the quadrupole term causes a Schwarzschild black hole to be prolate at the quadrupole strength Q<0 and oblate at Q>0,and the octopole term causes to the shadow stretch upward at the octopolar strength ngth O<0 and downward at O>0.The shadow of a Schwarzschild black hole with a halo stretches and squeezes along the horizontal direction at Q<0 and Q>0,respectively.Meanwhile,the black hole shadow shifts upward at O<0 and downward at O>0.We present the light rays that form the shadow boundary to explain the emergence of the extraordinary patterns of the black hole shadow with quadrupole and octopole terms.From the observable width W,height H,oblateness K,and distortion parameterδcof the black hole shadow,we can determine the quadrupole and octopolar strengths of a Schwarzschild black hole with a halo.Our results show that the quadrupolar and octopolar terms yield a series of interesting patterns for the shadow of a Schwarzschild black hole with a halo.

Systematic calculations of cluster radioactivity half-lives in trans-lead nuclei

In the present work,based on the Wentzel-Kramers-Brillouin(WKB)theory,considering the cluster preformation probability(Pc),we systematically investigate the cluster radioactivity half-lives of 22 trans-lead nuclei ranging from221Fr to242Cm.When the mass number of the emitted cluster Ac<28,Pcis obtained by the exponential relationship of Pcto theαdecay preformation probability(Pα)proposed by R.Blendowskeis et al.[Phys.Rev.Lett.61,1930(1988)],while Pαis calculated through the cluster-formation model(CFM).When Ac≥28,Pcis calculated through the charge-number dependence of Pcon the decay products proposed by Ren et al.[Phys.Rev.C70,034304(2004)].The half-lives of cluster radioactivity have been calculated by the density-dependent cluster model[Phys.Rev.C 70,034304(2004)]and by the unified formula of half-lives for alpha decay and cluster radioactivity[Phys.Rev.C 78,044310(2008)].For comparison,a universal decay law(UDL)proposed by Qi et al.[Phys.Rev.C 80,044326(2009)],a semi-empirical model for bothαdecay and cluster radioactivity proposed by Santhosh[J.Phys.G:Nucl.Part.Phys.35,085102(2008)],and a unified formula of half-lives for alpha decay and cluster radioactivity[Phys.Rev.C 78,044310(2008)]are also used.The calculated results of our work,Ni’s formula,and the UDL can well reproduce the experimental data and are better than those of Santhosh’s model.In addition,we extend this model to predict the half-lives for 51 nuclei,whose cluster radioactivity is energetically allowed or observed but not yet quantified in NUBASE2020.